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 $|K| > 10000^n\log R$. We show here a stronger bound, that in fact $|K| > (11 + o(1))^n\ln R$ suffices for arbitrary $1$-separated $K$, while the improvement $|K| > (5 + o(1))^n\ln R$ 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.
From: Gabriel Currier [view email]
[v1]
Mon, 15 Jun 2026 18:34:02 UTC (57 KB)
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