Chromatic numbers for the hyperbolic plane and discrete analogs

Abstract. We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at distance exactly d are of the same color. The problem depends on d and, following a strategy of Kloeckner, we show linear upper bounds on the necessary number of colors. In parallel, we study the same problem on q-regular trees and show analogous results. For both settings, we also consider a variant which consists in replacing d with an interval of distances.