Problem 188

Erdős Problem #188: declared status 'open'. Formalized: yes. 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 $1$? Current best: Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS75] proved $k\geq 5$. Tsaturian [Ts17] improved this to $k\geq 6$. Erd\H{o}s and Graham claim that $k\leq 10000000$ ('more or less'), but give no proof. Prize: no. Tags: geometry, ramsey theory.