On the chromatic numbers of spheres in ℝn

Abstract

Let χ(S n−1 r )) be the minimum number of colours needed to colour the points of a sphere S n−1 r of radius \(r \geqslant \tfrac{1} {2}\) in ℝn so that any two points at the distance 1 apart receive different colours. In 1981 P. Erdős conjectured that χ(S n−1 r )→∞ for all \(r \geqslant \tfrac{1} {2}\). This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S n−1 r ) ≥ n. In the same paper, Lovász claimed that if \(r < \sqrt {\frac{n} {{2n + 2}}}\), then χ(S n−1 r ) ≤ n+1, and he conjectured that χ(S n−1 r ) grows exponentially, provided \(r \geqslant \sqrt {\frac{n} {{2n + 2}}}\). In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S n−1 r ) grows exponentially for any \(r > \tfrac{1} {2}\).

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon, L. Babai, H. Suzuki: Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems, J. Comb. Th., Ser. A 58 (1991), 165–180.

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    L. Babai, P. Frankl: Linear algebra methods in combinatorics, Part 1, Department of Computer Science, The University of Chicago, Preliminary version 2, September 1992.

  3. [3]

    R. C. Baker, G. Harman, J. Pintz: The difference between consecutive primes, II, Proceedings of the London Mathematical Society, 83 (2001), 532–562.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    P. Brass, W. Moser, J. Pach: Research problems in discrete geometry, Springer, 2005.

  5. [5]

    H. Cramér: On the order of magnitude of the difference between consecutive prime numbers, Acta Arithmetica 2 (1936), 23–46.

    MATH  Google Scholar 

  6. [6]

    P. Erdős: Some unsolved problems, Magyar Tud. Akad. Mat. Kutató Int. Közl., 6 (1961), 221–254.

    Google Scholar 

  7. [7]

    P. Erdős, R. L. Graham: Problem proposed at the 6th Hungarian combinatorial conference, Eger, July 1981.

  8. [8]

    P. Frankl, R. M. Wilson: Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357–368.

    MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    A. B. Kupavskii: On colouring spheres in ℝn, Sbornik Mathematics 202(6) (2011), 83–110.

    MathSciNet  Article  Google Scholar 

  10. [10]

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

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    L. Lovász: Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Sci. Math. 45 (1983), 317–323.

    MATH  Google Scholar 

  12. [12]

    A. M. Raigorodskii: The Borsuk problem and the chromatic numbers of some metric spaces, Uspekhi Mat. Nauk 56(1) (2001), 107–146; English transl. in: Russian Math. Surveys 56(1) (2001), 103–139.

    MathSciNet  Google Scholar 

  13. [13]

    A. M. Raigorodskii: On the chromatic number of a space, Uspekhi Mat. Nauk 55(2) (2000), 147–148; English transl. in: Russian Math. Surveys 55(2) (2000), 351–352.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    A. M. Raigorodskii: The linear algebra method in combinatorics, Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, Russia, 2007 (in Russian).

    Google Scholar 

  15. [15]

    R. A. Rankin: The difference between consecutive prime numbers, V Proc. Edinburgh. Math. Soc. 13 (1962–1963), 331–332.

    MathSciNet  Article  Google Scholar 

  16. [16]

    C. A. Rogers: Covering a sphere with spheres, Mathematika 10 (1963), 157–164.

    MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    A. Schönhage: Eine Bemerkung zur Konstruktion grosser Primzahllucken, Archiv der Math. 14 (1963), 29–30.

    MATH  Article  Google Scholar 

  18. [18]

    G. J. Simmons: On a problem of Erdős concerning 3-colouring of the unit sphere, Discrete Math 8 (1974), 81–84.

    MathSciNet  MATH  Article  Google Scholar 

  19. [19]

    G. J. Simmons: The chromatic number of the sphere, J. Austral. Math. Soc. Ser 21 (1976), 473–480.

    MathSciNet  MATH  Article  Google Scholar 

  20. [20]

    A. Soifer: The Mathematical Coloring Book, Springer, 2009.

  21. [21]

    L. A. Székely: Erdős on unit distances and the Szemerédi-Trotter theorems, Paul Erdős and his Mathematics, Bolyai Series Budapest, J. Bolyai Math. Soc., Springer, 11 (2002), 649–666.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. M. Raigorodskii.

Additional information

This work is done under the financial support of the following grants: the grant 09-01-00294 of Russian Foundation for Basic Research, the grant MD-8390.2010.1 of the Russian President, the grant NSh-8784.2010.1 supporting Leading scientific schools of Russia, a grant of Dynastia foundation.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Raigorodskii, A.M. On the chromatic numbers of spheres in ℝn . Combinatorica 32, 111–123 (2012). https://doi.org/10.1007/s00493-012-2709-9

Download citation

Mathematics Subject Classification (2000)

  • 52C10
  • 05C15