The famous Hadwider-Nielson problem asks for the smallest number $n$ such that the points of the plane can be partitioned into $n$ sets such that no two points in the same set are at distance $1$.
This problem, among many similar problems, can be reformulated in terms of the chromatic number of certain Cayley graphs. In this talk, I will discuss two recent results in this direction: one pertains to the Borel chromatic number of the unit-distance graph associated to quadratic forms over local fields. The other result is a lower bound on the chromatic number of Cayley graphs of finite groups of Lie type.