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The chromatic number of space with forbidden regular simplex

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Abstract

An explicit exponentially growing lower bound for the chromatic number of Euclidean space with forbidden regular simplex is constructed.

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References

  1. A. Soifer, “The Chromatic Number of the Plane: Its Past, Present, and Future,” in Mat. Pros., Ser. 3 (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  MATH  MathSciNet  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  MATH  MathSciNet  Google Scholar 

  4. R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory (JohnWiley and Sons, New York, 1990).

    MATH  Google Scholar 

  5. P. Frankl and V. Rödl, “All triangles are Ramsey,” Trans. Amer. Math. Soc. 297 (2), 777–779 (1986).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Frankl and V. Rödl, “A partition property of simplices in Euclidean space,” J. Amer. Math. Soc. 3 (1), 1–7 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  8. 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  MATH  MathSciNet  Google Scholar 

  9. 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. Akad. Nauk 457 (2), 144–146 (2014) [Dokl. Math. 90, 432–434 (2014)].

    MATH  Google Scholar 

  10. 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)].

    Article  MATH  MathSciNet  Google Scholar 

  11. 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,” in Trudy Mat. Inst. Steklov., Vol. 288: Geometry, Topology, and Applications (2015), pp. 109–119 [Proc. Steklov Inst. Math. 288 (1), 94–104 (2015)].

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  13. A. M. Raigorodskii, “On distance graphs with large chromatic number but without large simplices,” Uspekhi Mat. Nauk 62 (6 (378)), 187–188 (2007) [RussianMath. Surveys 62 (6), 1224–1225 (2007)].

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  15. 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. 149–157.

    Chapter  Google Scholar 

  16. 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  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  18. A. E. Zvonarev and A. M. Raigorodskii, “On distance graphs with large chromatic and small clique numbers,” Trudy Mosk. Fiz.-Tekh. Inst. 4 (1), 122–126 (2012).

    Google Scholar 

  19. 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 

  20. E. E. Demekhin, A. M. Raigorodskii, and O. I. Rubanov, “Distance graphs having large chromatic numbers and not 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 

  21. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. D. D. Cherkashin and A. M. Raigorodskii, “On the chromatic number of low-dimensional spaces,” Dokl. Akad. Nauk 472 (1), 11–12 (2017) [Dokl. Math. 95 (1), 5–6 (2017)].

    MATH  MathSciNet  Google Scholar 

  24. A. M. Raigorodskii and D. V. Samirov, “New bounds for the chromatic number of a space with forbidden isosceles triangles,” Dokl. Akad. Nauk 456 (3), 280–283 (2014) [Dokl. Math. 89 (3), 313–316 (2014)].

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  26. P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry (Springer, New York, 2005).

    MATH  Google Scholar 

  27. A. Soifer, The Mathematical Coloring Book (Springer, New York, 2009).

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. A. M. Raigorodskii, “Around Borsuk’s hypothesis,” in Contemporary Mathematics. Fundamental Directions, Vol. 23: Geometry and Mechanics (2007), pp. 147–164 [J. Math. Sci. 154 (4), 604–623 (2008)].

    Google Scholar 

  30. A. M. Raigorodskii, “Coloring distance graphs and graphs of diameters,” in Thirty Essays on Geometric Graph Theory (Springer, New York, 2013), pp. 429–460.

    Chapter  Google Scholar 

  31. 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.

    MATH  MathSciNet  Google Scholar 

  32. K. Prachar, Primzahlverteilung (Springer-Verlag, Berlin, 1957; Mir, Moscow, 1967).

    MATH  Google Scholar 

  33. R. C. Baker, G. Harman, and J. Pintz, “The difference between consecutive primes. II,” Proc. LondonMath. Soc. (3) 83 (3), 532–562 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  34. P. Frankl and R. M. Wilson, “Intersection theorems with geometric consequences,” Combinatorica 1 (4), 357–368 (1981).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    A. A. Sagdeev

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, Russia

    A. A. Sagdeev

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  1. A. A. Sagdeev
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Correspondence to A. A. Sagdeev.

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Original Russian Text © A. A. Sagdeev, 2017, published in Matematicheskie Zametki, 2017, Vol. 102, No. 4, pp. 579–585.

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Sagdeev, A.A. The chromatic number of space with forbidden regular simplex. Math Notes 102, 541–546 (2017). https://doi.org/10.1134/S0001434617090267

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  • DOI: https://doi.org/10.1134/S0001434617090267

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