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

Combinatorics