On periodic sets avoiding given distance on the hyperbolic plane

We show that if a measurable set $I$ on the hyperbolic plane contains no pair of points at distance $r$ from each other, and is $\Gamma$-invariant for a finite covolume Fuchsian group $\Gamma\subset PSL_2(\mathbb{R})$, then it satisfies $\mu(\Gamma\backslash I) \leq (r+1)e^{-\frac{r}{2}} \mu(\Gamma\backslash\mathbb{H})$. As a corollary, we prove a lower bound for the measurable chromatic number of the graph on the points of a finite volume hyperbolic surface, where every two points at distance $r$ are connected by an edge. We show that for any finite volume hyperbolic surface this number is at least $e^{\frac{r}{2}}(r+1)^{-1}$.