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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 which 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 that our bounds are meaningful.