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Mat. Zametki, 2007, Volume 81, Issue 5, Pages 733–743 (Mi mz3717)  

This article is cited in 12 scientific papers (total in 12 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

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

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

Full text: PDF file (561 kB)
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English version:
Mathematical Notes, 2007, 81:5, 656–664

Bibliographic databases:

UDC: 519.174
Received: 06.07.2005
Revised: 18.08.2006

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

    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. 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  mathnet  crossref  mathscinet  elib
    4. A. M. Raigorodskii, D. V. Samirov, “Chromatic Numbers of Spaces with Forbidden Monochromatic Triangles”, Math. Notes, 93:1 (2013), 163–171  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. 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  mathnet  crossref  isi
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. A. V. Berdnikov, A. M. Raigorodskii, “On the Chromatic Number of Euclidean Space with Two Forbidden Distances”, Math. Notes, 96:5 (2014), 827–830  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. 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  crossref  mathscinet  zmath  isi  scopus
    9. 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  crossref  mathscinet  zmath  isi  scopus
    10. A. V. Berdnikov, “Estimate for the Chromatic Number of Euclidean Space with Several Forbidden Distances”, Math. Notes, 99:5 (2016), 774–778  mathnet  crossref  crossref  mathscinet  isi  elib
    11. E. S. Gorskaya, I. M. Mitricheva, “The chromatic number of the space $(\mathbb R^n, l_1)$”, Sb. Math., 209:10 (2018), 1445–1462  mathnet  crossref  crossref  isi  elib
    12. 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  mathnet  crossref  crossref  isi  elib
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