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arXiv · 1708.01081

Lower bounds for the measurable chromatic number of the hyperbolic plane

Evan DeCorte·Konstantin Golubev
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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.

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Evan DeCorte, Konstantin Golubev. 2017-08-03. Lower bounds for the measurable chromatic number of the hyperbolic plane. https://doi.org/10.1007/s00454-018-0027-8

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