Lower bounds for the measurable chromatic number of the hyperbolic plane
| Autor: | Konstantin Golubev, Evan DeCorte |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
050101 languages & linguistics
Hyperbolic geometry 05 social sciences Metric Geometry (math.MG) 02 engineering and technology Upper and lower bounds Graph Theoretical Computer Science Combinatorics Computational Theory and Mathematics Mathematics - Metric Geometry Euclidean geometry FOS: Mathematics 0202 electrical engineering electronic engineering information engineering Discrete Mathematics and Combinatorics Mathematics - Combinatorics 020201 artificial intelligence & image processing 0501 psychology and cognitive sciences Combinatorics (math.CO) Geometry and Topology Chromatic scale Spectral method 05C63 05C15 51M10 Mathematics |
| Popis: | 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. 13 pages, 2 figures |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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