The Chromatic Number of $\mathbb{R}^{n}$ with Multiple Forbidden Distances

Abstract

Let $A\subset\mathbb{R}_{>0}$ be a finite set of distances, and let $G_{A}(\mathbb{R}^{n})$ be the graph with vertex set $\mathbb{R}^{n}$ and edge set $\{(x,y)\in\mathbb{R}^{n}:\ \|x-y\|_{2}\in A\}$, and let $\chi(\mathbb{R}^{n},A)=\chi\left(G_{A}(\mathbb{R}^{n})\right)$. Erd\H{o}s asked about the growth rate of the $m$-distance chromatic number \[ \bar{\chi}(\mathbb{R}^{n};m)=\max_{|A|=m}\chi(\mathbb{R}^{n},A). \] We improve the best existing lower bound for $\bar{\chi}(\mathbb{R}^{n};m)$, and show that \[ \bar{\chi}(\mathbb{R}^{n};m)\geq\left(\Gamma_{\chi}\sqrt{m+1}+o(1)\right)^{n} \] where $\Gamma_{\chi}=0.79983\dots$ is an explicit constant. Our full result is more general, and applies to cliques in this graph. Let $\chi_{k}(G)$ denote the minimum number of colors needed to color $G$ so that no color contains a $(k+1)$-clique, and let $\bar{\chi}_{k}(\mathbb{R}^{n};m)$ denote the largest value this takes for any distance set of size $m$ . Using the Partition Rank Method, we show that \[ \bar{\chi}_{k}(\mathbb{R}^{n};m)>\left(\Gamma_{\chi}\sqrt{\frac{m+1}{k}}+o(1)\right)^{n}. \]