Given a finite group G acting freely on a compact metric space M, and \(\varepsilon >0\) , we define the G-Borsuk graph on M by drawing edges \(x\sim y\) whenever there is a non-identity \(g\in G\) such that \( \mathbb {d}\left( x,gy\right) \le \varepsilon \) . We show that when \(\varepsilon \) is small, its chromatic number is determined by the topology of M via its G-covering number, which is the minimum k such that there is a closed cover \(M=F_1\cup \dots \cup F_k\) with \(F_i\cap g(F_i)=\emptyset \) for all \(g\in G{\setminus }\{\mathbb {1}\}\) . We are interested in bounding this number. We give lower bounds using G-actions on Hom-complexes, and upper bounds using a recursive formula on the dimension of M. We conjecture that the true chromatic number coincides with the lower bound, and give computational evidence. We also study random G-Borsuk graphs, which are random induced subgraphs. For these, we compute thresholds for \(\varepsilon \) that guarantee that the chromatic number is still that of the whole G-Borsuk graph. Our results are tight (up to a constant) when the G-index and dimension of M coincide.
