Ramsey problems for graphs in Euclidean spaces and Cartesian powers

Abstract

Given a graph $H$, let $\chi_H(\mathbb{R}^n)$ be the smallest positive integer $r$ such that there exists an $r$-coloring of $\mathbb{R}^n$ with no monochromatic unit-copy of $H$, that is a set of $|V(H)|$ vertices of the same color such that any two vertices corresponding to an edge of $H$ are at distance one. This Ramsey-type function extends the famous Hadwiger--Nelson problem on the chromatic number $\chi(\mathbb{R}^n)=\chi_{K_2}(\mathbb{R}^n)$ of the space from a complete graph $K_2$ on two vertices to an arbitrary graph $H$. It also extends the classical Euclidean Ramsey problem for congruent monochromatic subsets to the family of those defined by a specific subset of unit distances. Among others, we show that $\chi_H(\mathbb{R}^n)=\chi(\mathbb{R}^n)$ for any even cycle $H$ of length $8$ or at least $12$ as well as for any forest and that $\chi_H(\mathbb{R}^n)=\lceil\chi(\mathbb{R}^n)/2\rceil$ for any sufficiently long odd cycle. Our main tools and results, which are of independent interest, establish that Cartesian powers enjoy Ramsey-type properties for graphs with favorable Tur\'an-type characteristics, such as zero hypercube Tur\'an density. In addition, we prove induced variants of these results, find bounds on $\chi_H(\mathbb{R}^n)$ for growing dimensions $n$, and prove a canonical-type result. We conclude with many open problems. One of these is to determine $\chi_{C_4}(\mathbb{R}^2)$, for a cycle $C_4$ on four vertices.