N. G. Moshchevitin, A. M. Raigorodskii, “Colorings of the Space $\mathbb R^n$ with Several Forbidden Distances”, Mat. Zametki, 81:5 (2007), 733–743; Math. Notes, 81:5 (2007), 656

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Matematicheskie Zametki, 2007, Volume 81, Issue 5, Pages 733–743
DOI: https://doi.org/10.4213/mzm3717 (Mi mzm3717)

This article is cited in 17 scientific papers (total in 17 papers)

Colorings of the Space

$\mathbb R^n$ with Several Forbidden Distances

N. G. Moshchevitin, A. M. Raigorodskii

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Full-text PDF (561 kB) Citations (17)

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DOI: https://doi.org/10.4213/mzm3717

Abstract: The paper is concerned with the classical problem concerning the chromatic number of a metric space, i.e., the minimal number of colors required to color all points in the space so that the distance (the value of the metric) between points of the same color does not belong to a given set of positive real numbers (the set of forbidden distances). New bounds for the chromatic number are obtained for the case in which the space is

$\mathbb R^n$ with a metric generated by some norm (in particular,

$l_p$ ) and the set of forbidden distances either is finite or forms a lacunary sequence.

Keywords: chromatic number, measurable chromatic number, coloring with forbidden distances, lacunary sequence, independence member of a graph, polyhedron, Diophantine approximation.

Received: 06.07.2005
Revised: 18.08.2006

English version:
Mathematical Notes, 2007, Volume 81, Issue 5, Pages 656–664
DOI: https://doi.org/10.1134/S0001434607050112

Bibliographic databases:

UDC: 519.174

Language: Russian

Citation: N. G. Moshchevitin, A. M. Raigorodskii, “Colorings of the Space

$\mathbb R^n$ with Several Forbidden Distances”, Mat. Zametki, 81:5 (2007), 733–743; Math. Notes, 81:5 (2007), 656–664

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/mzm3717

https://doi.org/10.4213/mzm3717

https://www.mathnet.ru/eng/mzm/v81/i5/p733

This publication is cited in the following 17 articles:

L. I. Bogolubsky, A. M. Raigorodskii, “A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics $\ell_1$ and $\ell_2$ ”, Math. Notes, 105:2 (2019), 180–203
E. S. Gorskaya, I. M. Mitricheva, “The chromatic number of the space $(\mathbb R^n, l_1)$ ”, Sb. Math., 209:10 (2018), 1445–1462
A. V. Berdnikov, “Estimate for the Chromatic Number of Euclidean Space with Several Forbidden Distances”, Math. Notes, 99:5 (2016), 774–778
A. V. Berdnikov, “Chromatic number with several forbidden distances in the space with the $\ell_q$ -metric”, Journal of Mathematical Sciences, 227:4 (2017), 395–401
Gil Kalai, Surveys in Combinatorics 2015, 2015, 147
S. N. Popova, “Zero-one law for random distance graphs with vertices in $\{-1,0,1\}^n$ ”, Problems Inform. Transmission, 50:1 (2014), 57–78
A. E. Zvonarev, A. M. Raigorodskii, D. V. Samirov, A. A. Kharlamova, “On the chromatic number of a space with forbidden equilateral triangle”, Sb. Math., 205:9 (2014), 1310–1333
A. V. Berdnikov, A. M. Raigorodskii, “On the Chromatic Number of Euclidean Space with Two Forbidden Distances”, Math. Notes, 96:5 (2014), 827–830
Zvonarev A.E., Raigorodskii A.M., Samirov D.V., Kharlamova A.A., “Improvement of the Frankl-Rodl Theorem on the Number of Edges in Hypergraphs With Forbidden Cardinalities of Edge Intersections”, Dokl. Math., 90:1 (2014), 432–434
Samirov D.V., Raigorodskii A.M., “New Bounds For the Chromatic Number of a Space With Forbidden Isosceles Triangles”, Dokl. Math., 89:3 (2014), 313–316
A. M. Raigorodskii, D. V. Samirov, “Chromatic Numbers of Spaces with Forbidden Monochromatic Triangles”, Math. Notes, 93:1 (2013), 163–171
D. V. Samirov, A. M. Raigorodskii, “New lower bounds for the chromatic number of a space with forbidden isosceles triangles”, J. Math. Sci. (N. Y.), 204:4 (2015), 531–541
Andrei M. Raigorodskii, Thirty Essays on Geometric Graph Theory, 2013, 429
N. G. Moshchevitin, “Density modulo 1 of lacunary and sublacunary sequences: application of Peres–Schlag's construction”, J. Math. Sci., 180:5 (2012), 610–625
A. M. Raigorodskii, “On the chromatic numbers of spheres in Euclidean spaces”, Dokl. Math., 81:3 (2010), 379
E. S. Gorskaya, I. M. Mitricheva (Shitova), V. Yu. Protasov, A. M. Raigorodskii, “Estimating the chromatic numbers of Euclidean space by convex minimization methods”, Sb. Math., 200:6 (2009), 783–801
A. M. Raigorodskii, I. M. Shitova, “Chromatic numbers of real and rational spaces with real or rational forbidden distances”, Sb. Math., 199:4 (2008), 579–612

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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