Problem 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 $1$ ?

Source: erdosproblems.com/188

Geometry
Ramsey theory

Wiki page

Status

Open

No claim settles this problem.

Open: the site labels the problem OPEN. The public sources compiled as of 2026-09-13 give

6\leq K_*\leq6330.

The lower bound is Tsaturian's refereed theorem (claim page (Tsaturian, 2017)). The upper bound is from the revised Currier–Mody–Xie–Zhang preprint (claim page (Currier, Mody, Xie and Zhang, 2026)); the best refereed upper bound is Conlon and Fox's $10^{10}$ (claim page (Conlon and Fox, 2017)). The site's commentary, last edited 14 October 2025, does not cite the Currier–Mody–Xie–Zhang bound. These are the compiled source-backed bounds, not the outcome of an exhaustive literature census.

ErGr79: P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics, L'Enseignement Mathématique (2) 25 (1979), 325–344; the question is on pp. 330–331.
ErGr80: P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique 28 (1980); the same question is on pp. 14–15. See the source record. - Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus, Euclidean Ramsey theorems. II (1975), 529–557. - Tsaturian, A Euclidean Ramsey Result in the Plane, Electronic Journal of Combinatorics 24(4) (2017), P4.35, doi:10.37236/7148. - Currier, Mody, Xie, and Zhang, Improved bounds for lines and $1$ -separated sets in Euclidean Ramsey theory, arXiv:2606.17194v2, August 31, 2026.

Problem 188

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Proof claims

Discussion