Improved bounds for lines and $1$-separated sets in Euclidean Ramsey theory

Abstract

Let $K$ be a $1$-separated set of diameter at most $R-1$, and let ${\ell }_m$ denote a collection of $m$ points on a line, with consecutive points of distance $1$ apart. Conlon and Fox (2019) demonstrated a coloring of $n$-dimensional Euclidean space avoiding red congruent copies of ${\ell }_2$ and blue congruent copies of $K$ for $|K|>10000^n{\log }_{2}R$. We show here a stronger bound, that in fact $|K|>(6.79+o(1){)}^n\log R$ suffices for arbitrary $1$-separated $K$, while the improvement $|K|>(5+o(1){)}^n\log R$ holds in many cases, including when $K={\ell }_m$, or more generally when $K$ is contained in a low-dimensional affine subspace. We also make a special study of the case when $n=2$, demonstrating a two-coloring of two-dimensional Euclidean space avoiding red copies of ${\ell }_2$ and blue copies of ${\ell }_{6330}$. This latter result addresses a question of Erdős and Graham.

13 pages, some improved results. Comments welcome!