Chromatic Numbers of Hyperbolic Surfaces

This article is about chromatic numbers of hyperbolic surfaces. For a metric space, the d-chromatic number is the minimum number of colors needed to color the points of the space so that any two points at distance d are of a different color. We prove upper bounds on the d-chromatic number of any hyperbolic surface that only depend on d. In another direction, we investigate chromatic numbers of closed genus g surfaces and find upper bounds that only depend on g (and not on d). For both problems, we construct families of examples that show our bounds are meaningful.