A notorious problem in combinatorics is the following:

If we color $\mathbb{R}^2$ so that no pair of points at unit distance get the same color, what is the fewest number of colors required?

This number is sometimes called "the chromatic number of the plane" $\chi$ (since it is really a graph coloring problem on an infinite graph), and it is easily see that $ 4 \le \chi \le 7$ but these bounds have stood for quite a while.

It is natural to ask about coloring other metric spaces, and many people have. In particular, there are bounds known for the chromatic number of $\mathbb{R}^d$ for $d \ge 3$ (and some asymptotics as $d \to \infty$). Also there has been some work on coloring the two-dimensional sphere of radius $r$. (A nice reference for this kind of problem is Soifer's "The mathematical coloring book.")

My question is whether anyone has looked at the chromatic number of the hyperbolic plane. As with the sphere, there is a free parameter --- one could either take fixed curvature $-1$ and let the distance vary, or fix unit distance and let the constant negative curvature vary.

We might not expect to be able to solve this in general, since determining the chromatic number of the plane seems difficult, and now we have an infinite family of such problems. But we should be able to put bounds, and I am wondering what is known.

In particular, the following two questions come to mind. I would appreciate insights or pointers to references if these things have been previously studied.

(1) Is there a $5$-chromatic unit distance graph in the hyperbolic plane (for some constant negative curvature)?

(2) Is there an absolute upper bound on the chromatic number of the hyperbolic plane, that holds for all constant negative curvatures?

One might guess that the chromatic number of the hyperbolic plane is increasing as the constant negative curvature decreases, but if so does it grow without bound?