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