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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On the chromatic number of an infinitesimal plane layer


Authors: A. Ya. Kanel-Belov, V. A. Voronov and D. D. Cherkashin
Translated by: D. D. Cherkashin
Original publication: Algebra i Analiz, tom 29 (2017), nomer 5.
Journal: St. Petersburg Math. J. 29 (2018), 761-775
MSC (2010): Primary 05C62, 52C10
DOI: https://doi.org/10.1090/spmj/1515
Published electronically: July 26, 2018
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Abstract | References | Similar Articles | Additional Information

Abstract: 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.


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Additional Information

A. Ya. Kanel-Belov
Affiliation: Moscow Institute of Physics and Technology, Lab of advanced combinatorics and network applications, Institutsky lane 9, Dolgoprudny, Moscow region, 141700, Russia
Email: kanelster@gmail.com

V. A. Voronov
Affiliation: Siberian Branch, Russian Academy of Sciences, Matrosov Institute for System Dynamics and Control Theory (ISDCT SB RAS), Lermontov Srt. 134, Irkutsk 664033, Russia
Email: v-vor@yandex.ru

D. D. Cherkashin
Affiliation: Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29B, St. Petersburg 199178, Russia; Moscow Institute of Physics and Technology, Lab of advanced combinatorics and network applications, Institutsky lane 9, Dolgoprudny, Moscow region 141700, Russia; St. Petersburg Branch V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia
Email: matelk@mail.ru

DOI: https://doi.org/10.1090/spmj/1515
Keywords: Chromatic number of the plane, chromatic number of Euclidean spaces
Received by editor(s): January 22, 2017
Published electronically: July 26, 2018
Additional Notes: The work was supported by the Russian Scientific Foundation: Theorems 7, 11 and Lemma 1 by the grant 16-11-10039; Theorems 8 and 9 by the grant 17-11-01377.
Article copyright: © Copyright 2018 American Mathematical Society

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