Journal of Mathematical Sciences

, Volume 165, Issue 6, pp 689–709 | Cite as

On independence numbers of distance graphs with vertices in {-1,0,1} n : estimates, conjectures, and applications to the Nelson–Erdős–Hadwiger problem and the Borsuk problem

  • A. E. Guterman
  • V. K. Lyubimov
  • A. M. Raigorodskii
  • S. A. Usachev


The paper states and studies a problem that is closely related to the problems mentioned in the title.


Small Dimension Chromatic Number Maximum Eigenvalue Distance Graph Independence Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    J. Abello, P. M. Pardalos, and M. G. C. Resende, “On maximum clique problems in very large graphs,”
  2. 2.
    P. K. Agarwal and J. Pach, Combinatorial Geometry, John Wiley and Sons, New York (1995).MATHGoogle Scholar
  3. 3.
    R. Ahlswede and V.M. Blinovsky, Lectures on Advances in Combinatorics, Springer (2008).Google Scholar
  4. 4.
    R. Ahlswede and L. H. Khachatrian, “The complete nontrivial-intersection theorem for systems of finite sets,” Combin. Theory, Ser. A, 76, 121–138 (1996).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. Ahlswede and L.H. Khachatrian, “The complete intersection theorem for systems of finite sets,” Eur. J. Combin., 18, 125–136 (1997).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    N. Alon and J. Spencer, The Probabilistic Method [Russian translation], Binom, Laboratoriya Znanii, Moscow (2007).Google Scholar
  7. 7.
    L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, Part 1, Department of Computer Science, The University of Chicago, Preliminary version 2, September 1992.Google Scholar
  8. 8.
    R. Bellman, Introduction to Matrix Theory [Russian translation], Nauka, Moscow (1969).Google Scholar
  9. 9.
    B. Bollob´as, Random Graphs, Second Edition, Cambridge Univ. Press (2001).Google Scholar
  10. 10.
    V. G. Boltyanskii and I. Ts. Gokhberg, Theorems and Problems of Combinatorial Geometry [in Russian], Nauka, Moscow (1965).Google Scholar
  11. 11.
    V. G. Boltyanskii, H. Martini, and P. S. Soltan, Excursions into Combinatorial Geometry, Universitext, Springer, Berlin (1997).Google Scholar
  12. 12.
    K. Borsuk, “Drei S¨atze ¨uber die n-dimensionale euklidische Sph¨are”, Fundam. Math., 20 , 177–190 (1933).Google Scholar
  13. 13.
    J. Bourgain and J. Lindenstrauss, “On covering a set in R d by balls of the same diameter,” In: Geometric Aspects of Functional Analysis, Lect. Notes Math., 1469, Springer-Verlag, Berlin (1991), pp. 138–144.CrossRefGoogle Scholar
  14. 14.
    P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry, Springer (2005).Google Scholar
  15. 15.
    N. G. de Bruijn and P. Erdős, “A colour problem for infinite graphs and a problem in the theory of relations,” Proc. Koninkl. Nederl. Acad. Wet., Ser. A, 54, No. 5, 371–373 (1951).MATHGoogle Scholar
  16. 16.
    J. Cibulka, “On the chromatic number of real and rational spaces,” Combinatorics, 18, No. 2, 53–66 (2008).MATHMathSciNetGoogle Scholar
  17. 17.
    D. Cvetkovicz, M. Doob, and H. Zachs, Spectra of Graphs: Theory and Applications [Russian translation], Naukova Dumka, Kiev (1984).Google Scholar
  18. 18.
    M. Deza and P. Frankl, “Erdős–Ko–Rado theorem – 22 years later,” SIAM J. Alg. Disc. Methods, 4, 419–431 (1983).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    P. Erdős, Ch. Ko, and R. Rado, “Intersection theorems for systems of finite sets,” J. Math. Oxford, Sec. 12 (48) (1961).Google Scholar
  20. 20.
    P. Frankl and V. Rϒodl, “Forbidden intersections,” Trans. Amer. Math. Soc., 300, No. 1, 259–286 (1987).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    P. Frankl and R. Wilson, “Intersection theorems with geometric consequences,” Combinatorics, 1, 357–368 (1981).MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    E. S. Gorskaya, V. Yu. Protasov, A. M. Raigorodskii, and I. M. Shitova, “Estimates for chromatic numbers of spaces via convex optimization methods,” Mat. Sb. (2009) (in press).Google Scholar
  23. 23.
    H. Hadwiger, “Ein ¨Uberdeckungssatz f¨ur den Euklidischen Raum,” Portug. Math., 4, 140–144 (1944).MATHMathSciNetGoogle Scholar
  24. 24.
    F. Harari, Graph Theory [Russian translation], Mir, Moscow (1973).Google Scholar
  25. 25.
    L. L. Ivanov, “Estimate for the chromatic number of space R4,” Usp. Mat. Nauk, 61, No. 5, 371–372 (2006).Google Scholar
  26. 26.
    J. Kahn and G. Kalai, “A counterexample to Borsuk’s conjecture,” Bull. Amer. Math. Soc., 29, No. 1, 60–62 (1993).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    R. Karp, Reducibility among Combinatorial Problems, Plenum Press, New York (1972).Google Scholar
  28. 28.
    V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, American Mathematical Society (1991).Google Scholar
  29. 29.
    D. G. Larman and C. A. Rogers, “The realization of distances within sets in Euclidean space,” Mathematika, 19, 1–24 (1972).MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    J. Matoušek, Using the Borsuk–Ulam Theorem, Universitext, Springer, Berlin (2003).MATHGoogle Scholar
  31. 31.
    B. Parlett, Symmetric Eigenvalue Problem. Numerical Methods [Russian translation], Mir, Moscow (1983)MATHGoogle Scholar
  32. 32.
    A. M. Raigorodskii, “The Borsuk problem and chromatic numbers of metric spaces,” Usp. Mat. Nauk, 56, No. 1, 107–146.Google Scholar
  33. 33.
    A. M. Raigorodskii, Linear-Algebraic Method in Combinatorics [in Russian], MTsNMO, Moscow (2007).Google Scholar
  34. 34.
    A. M. Raigorodskii, Chromatic Numbers [in Russian], MTsNMO, Moscow (2003).Google Scholar
  35. 35.
    L.A. Sz´ekely, “Erdős on unit distances and the Szemer´edi–Trotter theorems,” In: Paul Erdős and His Mathematics, Bolyai Series J. Bolyai Math. Soc., 11, Budapest, Springer (2002), pp. 649–666.Google Scholar
  36. 36.
    A. M. Raigorodskii, “On the Borsuk conjecture,” In: Progress in Science and Technology, Series on Contemporary Mathematics and Its Applications [in Russian], 23, All-Russian Institute for Scientific and Technical Information, Russian Academy of Sciences, Moscow (2007), pp. 147–164.Google Scholar
  37. 37.
    A. M. Raigorodskii, “Three lectures on the Borsuk partition problem,” In: London Mathematical Society Lecture Note Series, 347 (2007), pp. 202–248.Google Scholar
  38. 38.
    A. M. Raigorodskii, “The Borsuk partition problem: Its seventieth anniversary,” Math. Intell., 26, 4–12 (2004).MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    A. M. Raigorodskii, The Borsuk Problem [in Russian], MTsNMO, Moscow (2006).Google Scholar
  40. 40.
    A. M. Raigorodskii, “On lower bounds for Borsuk and Hadwiger numbers,” Usp. Mat Nauk, 59, No. 3, 177–178 (2004).MathSciNetGoogle Scholar
  41. 41.
    A. M. Raigorodskii, “On the relation between the Borsuk and Erdős–Hadwiger problems,” Usp. Mat. Nauk, 60, No. 4, 219–220 (2005).MathSciNetGoogle Scholar
  42. 42.
    A. M. Raigorodskii, “On the Borsuk and Erdős–Hadwiger problems,” Mat. Zametki, 79, No. 6, 913–924 (2006).MathSciNetGoogle Scholar
  43. 43.
    A. M. Raigorodskii, “On the chromatic number of space,” Usp. Mat. Nauk, 55, No. 2, 147–148 (2000).MathSciNetGoogle Scholar
  44. 44.
    A. M. Raigorodskii, Probability and Algebra in Combinatorics [in Russian], MTsNMO, Moscow (2008).Google Scholar
  45. 45.
    A. M. Raigorodskii, “Some problems in combinatorial geometry, and the linear algebra method in combinatorics,” Chebyshev Sb., 7, No. 3 (19), 168–189 (2006) .MATHMathSciNetGoogle Scholar
  46. 46.
    A. M. Raigorodskii, “The Borsuk and Hadwiger problems with forbidden inner products,” Usp. Mat. Nauk, 57, No. 3, 159–160 (2002).MathSciNetGoogle Scholar
  47. 47.
    A. M. Raigorodskii, “On dimension in the Borsuk problem,” Usp. Mat. Nauk, 52, No. 6, 181–182 (1997).MathSciNetGoogle Scholar
  48. 48.
    A. M. Raigorodskii, Systems of General Representatives in Combinatorics and Their Applications to Geometry, MTsNMO, Moscow (2009).Google Scholar
  49. 49.
    A. M. Raigorodskii and M. M. Kityaev, “On a series of problems related to Borsuk and Nelson–Erdős–Hadwiger problems,” Mat. Zametki, 84, No. 2, 254–272 (2008).MathSciNetGoogle Scholar
  50. 50.
    A. M. Raigorodskii and I. M. Shitova, “On the chromatic number of Euclidean space and the Borsuk problem,” Mat. Zametki, 83, No. 4, 636–639 (2008).MathSciNetGoogle Scholar
  51. 51.
    A. M. Raigorodskii and I. M Shitova, “On chromatic numbers of real and rational spaces with several real or several rational forbidden distances,” Mat. Sb., 199, No. 4, 107–142 (2008).MathSciNetGoogle Scholar
  52. 52.
    O. Schramm, “Illuminating sets of constant width,” Mathematika, 35, 180–189 (1988).MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    I. M. Shitova “On chromatic numbers of metric spaces with two forbidden distances,” Dokl. Ross. Akad. Nauk, 413, No. 2, 178–180 (2007).MathSciNetGoogle Scholar
  54. 54.
    A. Soifer, “Plane chromatic number; its past, present, and future,” Mat. Prosveshchenie, No. 8 (2004).Google Scholar
  55. 55.
    A. Soifer, “Chromatic number of the plane: a historical essay,” Combinatorics, 13–15 (1991).Google Scholar
  56. 56.
    V. E. Tarakanov, Combinatorial Problems and (0, 1)-Matrices [in Russian], Nauka, Moscow (1985).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • A. E. Guterman
    • 1
  • V. K. Lyubimov
    • 1
  • A. M. Raigorodskii
    • 1
  • S. A. Usachev
    • 1
  1. 1.Moscow State UniversityMoscowRussia

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