Skip to main content
SpringerLink
Log in

Lower bounds for the chromatic numbers of distance graphs with large girth

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We obtain some specific exponential lower bounds for the chromatic numbers of distance graphs with large girth.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Soifer, “Chromatic Number of the Plane: Its Past, Present, and Future,” in Matem. Prosv., Ser. 3 (Izd. MTsNMO, Moscow, 2004), Vol. 8, pp. 186–221 [in Russian].

    Google Scholar 

  2. A. M. Raigorodskii, “On the chromatic number of a space,” Uspekhi Mat. Nauk 55 (2 (332)), 147–148 (2000) [Russian Math. Surveys 55 (2), 351–352 (2000)].

    Article  MathSciNet  MATH  Google Scholar 

  3. D. G. Larman and C. A. Rogers, “The realization of distances within sets in Euclidean space,” Mathematika 19, 1–24 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  4. N. G. de Bruijn and P. Erdős, “A colour problem for infinite graphs and a problem in the theory of relations,” Nederl. Akad.Wet., Proc., Ser. A 54 (5), 371–373 (1951).

    MATH  Google Scholar 

  5. P. Erdős, “Graph theory and probability,” Canad. J.Math. 11, 34–38 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Erdős, “Unsolved problems,” in Congres Numerantium (UtilitasMath., Winnipeg, Man., 1976), Vol. XV, pp. 678–696.

    Google Scholar 

  7. P. O’Donnell, “Arbitrary girth, 4-chromatic unit distance graphs in the plane. I. Graph embedding,” Geombinatorics 9 (3), 145–150 (2000).

    MathSciNet  MATH  Google Scholar 

  8. P. O’Donnell, “Arbitrary girth, 4-chromatic unit distance graphs in the plane. II. Graph embedding,” Geombinatorics 9 (4), 180–193 (2000).

    MathSciNet  MATH  Google Scholar 

  9. A. B. Kupavskiy, “Distance graphs with large chromatic number and arbitrary girth,” Mosc. J. Comb. Number Theory 2 (2), 52–62 (2012).

    MathSciNet  Google Scholar 

  10. A. M. Raigorodskii, “On distance graphs that have a large chromatic number but do not contain large simplices,” Uspekhi Mat. Nauk 62 (6 (378)), 187–188 (2007) [Russian Math. Surveys 62 (6), 1224–1225 (2007)].

    Article  MathSciNet  Google Scholar 

  11. A. M. Raigorodskii and O. I. Rubanov, “Small clique and large chromatic number,” in European Conference on Combinatorics, Graph Theory and Applications, Electron. Notes Discrete Math. (Elsevier Sci. B. V., Amsterdam, 2009), Vol. 34, pp. 441–445.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. M. Raigorodskii and O. I. Rubanov, “On the clique and the chromatic numbers of high-dimensional distance graphs,” in Number Theory and Applications (Hindustan Book Agency, New Delhi, 2009), pp. 149–155.

    Google Scholar 

  13. A. M. Raigorodskii and O. I. Rubanov, “Distance graphs with large chromatic number and without large cliques,” Mat. Zametki 87 (3), 417–428 (2010) [Math. Notes 87 (3–4), 392–402 (2010)].

    Article  MathSciNet  MATH  Google Scholar 

  14. A. B. Kupavskii and A. M. Raigorodskii, “Distance graphs with large chromatic numbers and small clique numbers,” Dokl. Ross. Akad. Nauk 444 (5), 483–487 (2012) [Dokl.Math. 85 (3), 394–398 (2012)].

    MathSciNet  MATH  Google Scholar 

  15. A. E. Zvonarev and A. M. Raigorodskii, “Distance graphs with large chromatic numbers and small clique numbers,” Trudy MFTI 4 (1), 122–126 (2012).

    Google Scholar 

  16. A. B. Kupavskii and A. M. Raigorodskii, “Obstructions to the realization of distance graphs with large chromatic numbers on spheres of small radii,” Mat. Sb. 204 (10), 47–90 (2013) [Sb. Math. 204 (10), 1435–1479 (2013)].

    Article  MATH  Google Scholar 

  17. E. E. Demekhin, A. M. Raigorodskii, and O. I. Rubanov, “Distance graphs having large chromatic numbers and containing no cliques or cycles of a given size,” Mat. Sb. 204 (4), 49–78 (2013) [Sb. Math. 204 (4), 508–538 (2013)].

    Article  MATH  Google Scholar 

  18. A. B. Kupavskii, “Explicit and probabilistic constructions of distance graphs with small clique numbers and large chromatic numbers,” Izv. Ross. Akad. Nauk Ser.Mat. 78 (1), 65–98 (2014) [Izv. Math. 78 (1), 59–89 (2014)].

    Article  MathSciNet  MATH  Google Scholar 

  19. N. Alon and A. Kupavskii, “Two notions of unit distance graphs,” J. Combin. Theory Ser. A 125, 1–17 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  20. P. K. Agarwal and J. Pach, Combinatorial Geometry (JohnWiley and Sons, New York, 1995).

    MATH  Google Scholar 

  21. P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry (Springer-Verlag, Berlin, 2005).

    MATH  Google Scholar 

  22. A. Soifer, The Mathematical Coloring Book. Mathematics of Coloring and the Colorful Life of Its Creators (Springer-Verlag, Berlin, 2009).

    MATH  Google Scholar 

  23. A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56 (1 (337)), 107–146 (2001) [RussianMath. Surveys 56 (1), 103–139 (2001)].

    Article  MathSciNet  MATH  Google Scholar 

  24. A. M. Raigorodskii, “Around Borsuk’s hypothesis,” Sovrem. Mat., Fundam. Napravl. 23, 147–164 (2007) [J.Math. Sci. 154 (4), 604–623 (2008)].

    MATH  Google Scholar 

  25. A. M. Raigorodskii, “Coloring Distance Graphs and Graphs of Diameters,” in Thirty Essays on Geometric Graph Theory (Springer-Verlag, Berlin, 2013), pp. 429–460.

    Chapter  Google Scholar 

  26. A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters,” in Discrete Geometry and Algebraic Combinatorics, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2014), Vol. 625, pp. 93–109.

    MathSciNet  MATH  Google Scholar 

  27. P. Frankl and V. Rödl, “Forbidden intersections,” Trans. of Amer.Math. Soc. 300 (1), 259–286 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  28. A. E. Zvonarev, A.M. Raigorodskii, D.V. Samirov, and A. A. Kharlamova, “Improvement of the Frankl–Rödl theorem on the number of edges of a hypergraph with forbidden intersection,” Dokl. Ross. Akad. Nauk 457 (2), 144–146 (2014) [Dokl.Math. 90 (1), 432–434 (2014)].

    MATH  Google Scholar 

  29. A. E. Zvonarev, A. M. Raigorodskii, D. V. Samirov, and A. A. Kharlamova, “On the chromatic number of a space with forbidden equilateral triangle,” Mat. Sb. 205 (9), 97–120 (2014) [Sb. Math. 205 (9), 1310–1333 (2014)].

    Article  MathSciNet  MATH  Google Scholar 

  30. A. E. Zvonarev and A.M. Raigorodskii, “Improvements of the Frankl–Rödl theorem on the number of edges of a hypergraph with forbidden intersections, and their consequences in the problem of finding the chromatic number of a space with forbidden equilateral triangle,” Trudy Mat. Inst. Steklov 288, 109–119 (2015) [Proc. Steklov Inst. Math. 288, 94–104 (2015)]. Proc. Steklov Inst.Math. 288, 94-104 (2015); translation from Tr.Mat. Inst. Steklova 288, 109-119 (2015).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. “Mathematical Notes,” Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    A. A. Sagdeev

Authors
  1. A. A. Sagdeev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Sagdeev.

Additional information

Original Russian Text © A. A. Sagdeev, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 3, pp. 430–445.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sagdeev, A.A. Lower bounds for the chromatic numbers of distance graphs with large girth. Math Notes 101, 515–528 (2017). https://doi.org/10.1134/S0001434617030130

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434617030130

Keywords

Search

Navigation