Problems of Information Transmission

, Volume 54, Issue 1, pp 70–83 | Cite as

Chromatic Numbers of Distance Graphs with Several Forbidden Distances and without Cliques of a Given Size

Large Systems


We consider distance graphs with k forbidden distances in an n-dimensional space with the p-norm that do not contain cliques of a fixed size. Using a probabilistic construction, we present graphs of this kind with chromatic number at least (Bk) Cn , where B and C are constants.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    de Bruijn, N.G. and Erdős, P., A Colour Problem for Infinite Graphs and a Problem in the Theory of Relations, Nederl. Akad. Wetensch. Proc. Ser. A, 1951, vol. 54, no. 5, pp. 374–382.CrossRefMATHGoogle Scholar
  2. 2.
    Raigorodskii, A.M., On the Chromatic Number of a Space, Uspekhi Mat. Nauk, 2000, vol. 55, no. 2, pp. 147–148 [Russian Math. Surveys (Engl. Transl.), 2000, vol. 55, no. 2, pp. 351–352].MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Frankl, P. and Wilson, R.M., Intersection Theorems with Geometric Consequences, Combinatorica, 1981, vol. 1, no. 4, pp. 357–368.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Larman, D.G. and Rogers, C.A., The Realization of Distances within Sets in Euclidean Space, Mathematika, 1972, vol. 19, no. 1, pp. 1–24.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Füredi, Z. and Kang, J.-H., Distance Graph on Zn with ł1 Norm, Theoret. Comput. Sci., 2004, vol. 319, no. 1–3, pp. 357–366.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Raigorodskii, A.M., On the Chromatic Number of a Space with the Metric łq, Uspekhi Mat. Nauk, 2004, vol. 59, no. 5 (359), pp. 161–162 [Russian Math. Surveys (Engl. Transl.), 2004, vol. 59, no. 5, pp. 973–975].MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kupavskiy, A., On the Chromatic Number of Rn with an Arbitrary Norm, Discrete Math., 2011, vol. 311, no. 6, pp. 437–440.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gorskaya, E.S., Mitricheva, I.M., Protasov, V.Yu., and Raigorodskii, A.M., Estimating the Chromatic Numbers of Euclidean Space by Convex Minimization Methods, Mat. Sb., 2009, vol. 200, no. 6, pp. 3–22 [Sb. Math. (Engl. Transl.), 2009, vol. 200, no. 5–6, pp. 783–801].CrossRefMATHGoogle Scholar
  9. 9.
    Berdnikov, A.V., Chromatic Number with Several Forbidden Distances in the Space with the łq-Metric, Sovrem. Matem. Prilozh., 2016, vol. 100, pp. 12–18 [J. Math. Sci. (N.Y.) (Engl. Transl.), 2017, vol. 227, no. 4, pp. 395–401].MATHGoogle Scholar
  10. 10.
    Berdnikov, A.V., Estimate for the Chromatic Number of Euclidean Space with Several Forbidden Distances, Mat. Zametki, 2016, vol. 99, no. 5, pp. 783–787 [Math. Notes (Engl. Transl.), 2016, vol. 99, no. 5–6, pp. 774–778].MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Raigorodskii, A.M., On Distance Graphs That Have a Large Chromatic Number but Do Not Contain Large Simplices, Uspekhi Mat. Nauk, 2007, vol. 62, no. 6 (378) pp. 187–188 [Russian Math. Surveys (Engl. Transl.), 2007, vol. 62, no. 6, pp. 1224–1225].MathSciNetCrossRefGoogle Scholar
  12. 12.
    Raigorodskii, A.M. and Rubanov, O.I., Small Clique and Large Chromatic Number, Electron. Notes Discrete Math., 2009, vol. 34, pp. 441–445.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Raigorodskii, A.M. and Rubanov, O.I., On the Clique and the Chromatic Numbers of High-Dimensional Distance Graphs, Number Theory and Applications: Proceedings of the International Conferences on Number Theory and Cryptography, Adhikari, S.D. and Ramakrishna, B., Eds., Gurgaon, New Delhi: Hindustan Book Agency, 2009, pp. 149–157.CrossRefGoogle Scholar
  14. 14.
    Raigorodskii, A.M. and Rubanov, O.I., Distance Graphs with Large Chromatic Number and without Large Cliques, Mat. Zametki, 2010, vol. 87, no. 3, pp. 417–428 [Math. Notes (Engl. Transl.), 2010, vol. 87, no. 3–4, pp. 392–402].MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Demëkhin, E.E., Raigorodskii, A.M., and Rubanov, O.I., Distance Graphs That Have a Large Chromatic Number and Contain No Cliques or Cycles of a Given Size, Mat. Sb., 2013, vol. 204, no. 4, pp. 49–78 [Sb. Math. (Engl. Transl.), 2013, vol. 204, no. 3–4, pp. 508–538].MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kupavskii, A.B. and Raigorodskii, A.M., On Distance Graphs with Large Chromatic Numbers and Small Clique Numbers, Dokl. Akad. Nauk, 2012, vol. 444, no. 5, pp. 483–487 [Dokl. Math. (Engl. Transl.), 2012, vol. 85, no. 3, pp. 394–398].MathSciNetMATHGoogle Scholar
  17. 17.
    Kupavskii, A.B., Explicit and Probabilistic Constructions of Distance Graphs with Small Clique Numbers and Large Chromatic Numbers, Izv. Ross. Akad. Nauk, Ser. Mat., 2014, vol. 78, no. 1, pp. 65–98 [Izv. Math. (Engl. Transl.), 2014, vol. 78, no. 1, pp. 59–89].MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Erdős, P., Graph Theory and Probability, Canad. J. Math., 1959, vol. 11, pp. 34–38.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.MATHGoogle Scholar
  20. 20.
    Bassalygo, L., Cohen, G., and Zémor, G., Codes with Forbidden Distances, Discrete Math., 2000, vol. 213, no. 1–3, pp. 3–11.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Alon, N. and Spencer, J.H., The Probabilistic Method, New York: Wiley, 2000, 2nd ed. Translated under the title Veroyatnostnyi metod, Moscow: BINOM, 2011, 2nd ed.CrossRefMATHGoogle Scholar
  22. 22.
    Raigorodskii, A.M., Lineino-algebraicheskii metod v kombinatorike (The Linear Algebra Method in Combinatorics), Moscow: MCCME, 2015, 2nd ed.Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.Moscow Institute of Physics and Technology (State University)MoscowRussia

Personalised recommendations