Lower bounds for the measurable chromatic number of the hyperbolic plane

TL;DR

This paper establishes a lower bound of at least 6 colours for measurable colourings of the hyperbolic plane graph at large distances, extending the Euclidean chromatic problem to hyperbolic geometry.

Contribution

It introduces spectral methods to prove a new lower bound on the measurable chromatic number of hyperbolic plane graphs for large distances.

Findings

Lower bound of 6 colours for large d

Spectral methods applied to hyperbolic chromatic problems

Extension of Euclidean chromatic bounds to hyperbolic geometry

Abstract

Consider the graph $\mathbb{H}(d)$ whose vertex set is the hyperbolic plane, where two points are connected with an edge when their distance is equal to some $d>0$. Asking for the chromatic number of this graph is the hyperbolic analogue to the famous Hadwiger-Nelson problem about colouring the points of the Euclidean plane so that points at distance $1$ receive different colours. As in the Euclidean case, one can lower bound the chromatic number of $\mathbb{H}(d)$ by $4$ for all $d$. Using spectral methods, we prove that if the colour classes are measurable, then at least $6$ colours are are needed to properly colour $\mathbb{H}(d)$ when $d$ is sufficiently large.