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Mathematical Notes

, Volume 103, Issue 1–2, pp 243–250 | Cite as

Upper Bounds for the Chromatic Numbers of Euclidean Spaces with Forbidden Ramsey Sets

  • R. I. Prosanov
Article
  • 13 Downloads

Abstract

The chromatic number of a Euclidean space ℝ n with a forbidden finite set C of points is the least number of colors required to color the points of this space so that no monochromatic set is congruent to C. New upper bounds for this quantity are found.

Keywords

Euclidean Ramsey theory chromatic number of space 

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References

  1. 1.
    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)].MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A. Soifer, The Mathematical Coloring Book. Mathematics of Coloring and the Colorful Life of Its Creators (Springer, New York, 2009).MATHGoogle Scholar
  3. 3.
    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.Google Scholar
  4. 4.
    A. M. Raigorodskii, “Coloring distance graphs and graphs of diameters,” in Thirty Essays on Geometric Graph Theory (Springer, New York, 2013), pp. 429–460.CrossRefGoogle Scholar
  5. 5.
    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)].MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    D. G. Larman and C. A. Rogers, “The realization of distances within sets in Euclidean space,” Mathematika 19, 1–24 (1972).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    P. Erdos, R. L. Graham, P. Montgomery, B. L. Rothschild, J. H. Spencer, and E. G. Straus, “Euclidean Ramsey theorems,” J. Combin. Theory Ser. A 14, 341–363 (1973).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    P. Frankl and V. Rödl, “All triangles are Ramsey,” Trans. Amer. Math. Soc. 297 (2), 777–779 (1980).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    P. Frankl and V. Rödl, “A partition property of simplices in Euclidean space,” J. Amer. Math. Soc. 3 (1), 1–7 (1990).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    I. Kríž, “Permutation groups in EuclideanRamsey theory,” Proc. Amer. Math. Soc. 112 (3), 899–907 (1991).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    K. Cantwell, “All regular polytopes are Ramsey,” J. Combin. Theory Ser. A 114 (3), 555–562 (2007).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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 intersections,” Dokl. Ross. Akad. Nauk 457 (2), 144–146 (2014) [Dokl. Math. 90 (1), 432–434 (2014)].MATHGoogle Scholar
  13. 13.
    A. E. Zvonarev, A. M. Raigorodskii, D. V. Samirov, and A. A. Kharlamova, “O chromatic number space with forbiddenprohibited equilateral triangle,” Mat. Sb. 205 (9), 97–120 (2014) [Sb. Math. 205 (9), 1310–1333 (2014)].MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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, Vol. 288: Geometry, Topology and Application (Nauka, Moscow, 2015), pp. 109–119 [Proc. Steklov Inst. Math. 288 (1), 94–104 (2015)].Google Scholar
  15. 15.
    A. M. Raigorodskii and A. A. Sagdeev, “On the chromatic number of a space with a forbidden regular simplex,” Dokl. Ross. Akad. Nauk 472 (2), 127–129 (2017) [Dokl. Math. 95 (1), 15–16 (2017)].MathSciNetMATHGoogle Scholar
  16. 16.
    A. A. Sagdeev, “Lower bounds for the chromatic numbers of distance graphs with large girth,” Mat. Zametki 101 (3), 430–445 (2017) [Math. Notes 101 (3–4), 515–528 (2017)].MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    A. A. Sagdeev, “On the chromatic number of space with a forbidden regular simplex,” Mat. Zametki (in press).Google Scholar
  18. 18.
    A. A. Sagdeev, “On a Frank–Rödl theorem and its geometric corollaries,” Electron. Notes in DiscreteMath. (in press).Google Scholar
  19. 19.
    H. Jung, “Über die kleinste Kugel, die eine räumliche Figur einschließt,” J. Reine Angew. Math 123, 241–257 (1901).MathSciNetMATHGoogle Scholar
  20. 20.
    G. J. Butler, “Simultaneous packing and covering in Euclidean space,” Proc. London Math. Soc. (3) 25, 721–735 (1972).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    P. Erdös and C. A. Rogers, “Covering space with convex bodies,” Acta Arith. 7, 281–285 (1962).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    M. Naszodi, “On some covering problems in geometry,” Proc. Amer. Math. Soc. 144 (8), 3555–3562 (2016).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    A. B. Kupavskii, “On the chromatic number of Rn with an arbitrary norm,” DiscreteMath. 311 (6), 437–440 (2011).Google Scholar
  24. 24.
    L. Lovász, “On the ratio of optimal integral and fractional covers,” DiscreteMath. 13 (4), 383–390 (1975).MathSciNetMATHGoogle Scholar
  25. 25.
    Z. Füredi and J.-H. Kang, “Covering the n-space by convex bodies and its chromatic number,” Discrete Math. 308 (19), 4495–4500 (2008).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    A. B. Kupavskii, “On the chromatic number of Rn with a set of forbidden distances,” Dokl. Ross. Akad. Nauk 435 (6), 740–743 (2010) [Dokl. Math. 82 (3), 963–966 (2010)].Google Scholar
  27. 27.
    R. Prosanov, A New Proof of the Larman–Rogers Upper Bound for the Chromatic Number of the Euclidean Space, 1610.02846 (2016).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • R. I. Prosanov
    • 1
  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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