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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 1, Pages 65–98 (Mi izv7962)  

This article is cited in 11 scientific papers (total in 11 papers)

Explicit and probabilistic constructions of distance graphs with small clique numbers and large chromatic numbers

A. B. Kupavskii

Moscow Institute of Physics and Technology

Abstract: We study distance graphs with exponentially large chromatic numbers and without $k$-cliques, that is, complete subgraphs of size $k$. Explicit constructions of such graphs use vectors in the integer lattice. For a large class of graphs we find a sharp threshold for containing a $k$-clique. This enables us to improve the lower bounds for the maximum of the chromatic numbers of such graphs. We give a new probabilistic approach to the construction of distance graphs without $k$-cliques, and this yields better lower bounds for the maximum of the chromatic numbers for large $k$.

Keywords: distance graph, chromatic number, clique number, Nelson problem.

DOI: https://doi.org/10.4213/im7962

Full text: PDF file (692 kB)
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English version:
Izvestiya: Mathematics, 2014, 78:1, 59–89

Bibliographic databases:

UDC: 519.174.7+519.176+519.175.4
MSC: 52C10, 05C10, 05C15, 05C69, 05D10, 05C50
Received: 08.02.2012
Revised: 12.07.2012

Citation: A. B. Kupavskii, “Explicit and probabilistic constructions of distance graphs with small clique numbers and large chromatic numbers”, Izv. RAN. Ser. Mat., 78:1 (2014), 65–98; Izv. Math., 78:1 (2014), 59–89

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters”, Discrete Geometry and Algebraic Combinatorics, Contemporary Mathematics, 625, 2014, 93–109  crossref  mathscinet  zmath  isi
    2. A. Sagdeev, “Lower Bounds for the Chromatic Numbers of Distance Graphs with Large Girth”, Math. Notes, 101:3 (2017), 515–528  mathnet  crossref  crossref  mathscinet  isi  elib
    3. A. Sagdeev, “The Chromatic Number of Space with Forbidden Regular Simplex”, Math. Notes, 102:4 (2017), 541–546  mathnet  crossref  crossref  mathscinet  isi  elib
    4. A. Sokolov, “On the Chromatic Numbers of Rational Spaces”, Math. Notes, 103:1-2 (2018), 111–117  mathnet  crossref  crossref  mathscinet  isi  elib
    5. P. Frankl, A. Kupayskii, “Erdös-Ko-Rado theorem for $\{0,\pm 1\}$-vectors”, J. Comb. Theory Ser. A, 155 (2018), 157–179  crossref  mathscinet  zmath  isi  scopus
    6. P. Frankl, “An exact result for $(0,\pm 1)$-vectors”, Optim. Lett., 12:5 (2018), 1011–1017  crossref  mathscinet  zmath  isi  scopus
    7. P. Frankl, A. Kupavskii, “Families of vectors without antipodal pairs”, Stud. Sci. Math. Hung., 55:2 (2018), 231–237  crossref  mathscinet  zmath  isi  scopus
    8. A. V. Berdnikov, “Chromatic numbers of distance graphs with several forbidden distances and without cliques of a given size”, Problems Inform. Transmission, 54:1 (2018), 70–83  mathnet  crossref  isi  elib
    9. R. I. Prosanov, “Counterexamples to Borsuk's Conjecture with Large Girth”, Math. Notes, 105:6 (2019), 874–880  mathnet  crossref  crossref  isi  elib
    10. Yu. A. Demidovich, “Distance Graphs with Large Chromatic Number and without Cliques of Given Size in the Rational Space”, Math. Notes, 106:1 (2019), 38–51  mathnet  crossref  crossref  isi  elib
    11. Frankl P., Kupavskii A., “Two Problems on Matchings in Set Families - in the Footsteps of Erdos and Kleitman”, J. Comb. Theory Ser. B, 138 (2019), 286–313  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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