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

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 \(\chi(\mathbb{R}^{n},A)=\chi\left(G_{A}(\mathbb{R}^{n})\right)\). Erdős 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 \[ \bar{\chi}_{k}(\mathbb{R}^{n};m)>\left(\Gamma_{\chi}\sqrt{\frac{m+1}{k}}+o(1)\right)^{n}. \]