The chromatic number of Rn$\mathbb {R}^{n}$ with multiple forbidden distances

Abstract

AbstractLet be a finite set of distances, and let be the graph with vertex set and edge set , and let . Erdős asked about the growth rate of the m‐distance chromatic number We improve the best existing lower bound for , and show that where is an explicit constant. Our full result is more general, and applies to cliques in this graph. Let denote the minimum number of colors needed to color G so that no color contains a ‐clique, and let denote the largest value this takes for any distance set of size m. Using the partition rank method, we show that