On the chromatic number of an infinitesimal plane layer

This paper is devoted to a natural generalization of the problem on the chromatic number of the plane. The chromatic number of the spaces \mathbb{R}^n \times [0,\varepsilon ]^k is considered. It is proved that 5 \leq \chi (\mathbb{R}^2\times [0,\varepsilon ])\leq 7 and {6\leq \chi (\mathbb{R}^2\time...

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Bibliographic Details
Published in:St. Petersburg mathematical journal 2018-01, Vol.29 (5), p.761-775
Main Authors: Kanel-Belov, A. Ya, Voronov, V. A., Cherkashin, D. D.
Format: Article
Language:English
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Summary:This paper is devoted to a natural generalization of the problem on the chromatic number of the plane. The chromatic number of the spaces \mathbb{R}^n \times [0,\varepsilon ]^k is considered. It is proved that 5 \leq \chi (\mathbb{R}^2\times [0,\varepsilon ])\leq 7 and {6\leq \chi (\mathbb{R}^2\times [0,\varepsilon ]^2) \leq 7} for \varepsilon >0 sufficiently small. Also, some natural questions arising from these considerations are posed.
ISSN:1061-0022
1547-7371
DOI:10.1090/spmj/1515