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 $χ(\mathbb{R}^{n},A)=χ\left(G_{A}(\mathbb{R}^{n})\right)$. Erdős asked about the growth rate of the $m$-distance chromatic number \[ \barχ(\mathbb{R}^{n};m)=\max_{|A|=m}χ(\mathbb{R}^{n},A). \] We improve the best existing lower bound for $\barχ(\mathbb{R}^{n};m)$, and show that \[ \barχ(\mathbb{R}^{n};m)\geq\left(Γ_χ\sqrt{m+1}+o(1)\right)^{n} \] where $Γ_χ=0.79983\dots$ is an explicit constant. Our full result is more general, and applies to cliques in this graph. Let $χ_{k}(G)$ denote the minimum number of colors needed to color $G$ so that no color contains a $(k+1)$-clique, and let $\barχ_{k}(\mathbb{R}^{n};m)$ denote the largest value this takes for any distance set of size $m$ . Using the Partition Rank Method, we show that \[ \barχ_{k}(\mathbb{R}^{n};m)>\left(Γ_χ\sqrt{\frac{m+1}{k}}+o(1)\right)^{n}. \]