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 six colours are needed to properly colour \(\mathbb {H}(d)\) when d is sufficiently large.
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Acknowledgements
The authors would like to thank Shimon Brooks, Alexander Lubotzky, and Fernando Oliveira for helpful comments and remarks, Hillel Furstenberg for pointing out that a version of Lemma 3.2 ought to be true, and the anonymous referees for their comments which helped to improve the paper. This work was started when the first author held the position of Lady Davis Postdoctoral Fellow at the Hebrew University of Jerusalem, and the second author was a Ph.D. student there.
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Editor in Charge: Kenneth Clarkson
Evan DeCorte is supported by the CRM Applied Math Laboratory and NSERC Discovery Grant 2015-06746. During the first part of the work he was supported by ERC Grant GA 320924-ProGeoCom and the Lady Davis Fellowship Trust. Konstantin Golubev is supported by the ERC Grant 336283.
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DeCorte, E., Golubev, K. Lower Bounds for the Measurable Chromatic Number of the Hyperbolic Plane. Discrete Comput Geom 62, 481–496 (2019). https://doi.org/10.1007/s00454-018-0027-8
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DOI: https://doi.org/10.1007/s00454-018-0027-8
Keywords
- Measurable chromatic number
- Fourier methods
- Hyperbolic geometry
- Hoffman bound
- Infinite graph
- Hadwiger–Nelson problem