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Coloring Distance Graphs and Graphs of Diameters

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Thirty Essays on Geometric Graph Theory

Abstract

In this chapter, we discuss two classical problems lying on the edge of graph theory and combinatorial geometry. The first problem is due to E. Nelson. It consists of coloring metric spaces in such a way that pairs of points at some prescribed distances receive different colors. The second problem is attributed to K. Borsuk and involves finding the minimum number of parts of smaller diameter into which an arbitrary bounded nonsingleton point set in a metric space can be partitioned. Both problems are easily translated into the language of graph theory, provided we consider, instead of the whole space, any (finite) distance graph or any (finite) graph of diameters. During the last decades, a huge number of ideas have been proposed for solving both problems, and many results in both directions have been obtained. In the survey below, we try to give an entire picture of this beautiful area of geometric combinatorics.

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References

  1. A.B. Kupavskii, A.M. Raigorodskii, Distance Graphs with Large Chromatic Numbers and Small Clique Numbers, Doklady Mathematics, 85 No. 3, 394-398 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. A.B. Kupavskii, Distance graphs with large chromatic number and arbitrary girth, Moscow J. of Comb. and Number Theory, 2(N2), 52–62 (2012)

    MathSciNet  Google Scholar 

  3. P.K. Agarwal, J. Pach, Combinatorial Geometry (Wiley, New York, 1995)

    MATH  Google Scholar 

  4. R. Ahlswede, V.M. Blinovsky, Lectures on Advances in Combinatorics (Springer, Berlin, 2008)

    Book  MATH  Google Scholar 

  5. R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets. J. Combin. Theor. A 76, 121–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Ahlswede, L.H. Khachatrian, The complete intersection theorem for systems of finite sets. Eur. J. Combin. 18, 125–136 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Alon, J. Spencer, The Probabilistic Method, 2nd edn. (Wiley–Interscience Series in Discrete Mathematics and Optimization, New York, 2000)

    Google Scholar 

  9. L. Babai, P. Frankl, in Linear Algebra Methods in Combinatorics, Part 1, Department of Computer Science, University of Chicago, Preliminary version 2, September 1992

    Google Scholar 

  10. R.C. Baker, G. Harman, J. Pintz, The difference between consecutive primes, II. Proc. Lond. Math. Soc. 83, 532–562 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Benda, M. Perles, Colorings of metric spaces. Geombinatorics 9, 113–126 (2000)

    MathSciNet  MATH  Google Scholar 

  12. B. Bollobás, Random Graphs, 2nd edn. (Cambridge University Press, Cambridge, 2001)

    Book  MATH  Google Scholar 

  13. V.G. Boltyanski, H. Martini, P.S. Soltan, Excursions into Combinatorial Geometry, Universitext (Springer-Verlag, Berlin, 1997)

    Google Scholar 

  14. K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre. Fundamenta Math. 20, 177–190 (1933)

    Google Scholar 

  15. J. Bourgain, J. Lindenstrauss, On covering a set in \({\mathbb{R}}^{d}\) by balls of the same diameter, ed. by J. Lindenstrauss, V. Milman. Geometric Aspects of Functional Analysis, Lecture Notes in Math., vol. 1469 (Springer-Verlag, Berlin, 1991), pp. 138–144

    Google Scholar 

  16. P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry (Springer, Berlin, 2005)

    MATH  Google Scholar 

  17. N.G. de Bruijn, P. Erdős, A colour problem for infinite graphs and a problem in the theory of relations. Proc. Koninkl. Nederl. Acad. Wet., Ser. A 54(N5), 371–373 (1951)

    Google Scholar 

  18. K. Cantwell, Finite Euclidean Ramsey theory. J. Combin. Theor. A 73(N2), 273–285 (1996)

    MathSciNet  MATH  Google Scholar 

  19. K.B. Chilakamarri, The unit-distance graph problem: A brief survey and some new results. Bull. Inst. Comb. Appl. 8, 39–60 (1993)

    MathSciNet  MATH  Google Scholar 

  20. J. Cibulka, On the chromatic number of real and rational spaces. Geombinatorics 18(N2), 53-66 (2008)

    MathSciNet  MATH  Google Scholar 

  21. J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups (Springer-Verlag, Berlin, 1988)

    Book  MATH  Google Scholar 

  22. D. Coulson, A 15-coloring of 3-space omitting distance one. Discrete Math. 256, 83–90 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. H.T. Croft, Incident incidents. Eureka (Cambridge) 30, 22–26 (1967)

    Google Scholar 

  24. E.E. Demekhin, A.M. Raigorodskii, O.I. Rubanov, Distance graphs having large chromatic numbers and not containing cliques or cycles of a given size, to appear in Sb. Math.

    Google Scholar 

  25. P. O’Donnel, R. Hochberg, Some 4-chromatic unit-distance graphs without small cycles. Geombinatorics 5(N4), 137–142 (1996)

    MathSciNet  Google Scholar 

  26. P. O’Donnell, Arbitrary girth, 4-chromatic unit distnace graphs in the plane. Graph description. Geombinatorics 9, 145–152 (2000)

    MathSciNet  MATH  Google Scholar 

  27. P. O’Donnell, Arbitrary girth, 4-chromatic unit distance graphs in the plane. Graph embedding. Geombinatorics 9, 180–193 (2000)

    MathSciNet  MATH  Google Scholar 

  28. H.G. Eggleston, Covering a three-dimensional set with sets of smaller diameter. J. Lond. Math. Soc. 30, 11–24 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Erdős, in Unsolved Problems. Congress Numerantium XV—Proceedings of the 5th British Comb. Conf. 1975, (1976), p. 681

    Google Scholar 

  30. P. Erdős, On sets of distances of n points. Am. Math. Monthly 53, 248–250 (1946)

    Article  Google Scholar 

  31. P. Erdős, R.L. Graham, in Problem Proposed at the 6th Hungarian Combinatorial Conference, Eger, Hungary, July 1981

    Google Scholar 

  32. P. Erdős, C. Ko, R. Rado, Intersection theorems for systems of finite sets. J. Math. Oxford, Sec. 12(48) (1961)

    Google Scholar 

  33. P. Erdős, A. Rényi, On random graphs I. Publ. Math. Debrecen 6, 290–297 (1959)

    MathSciNet  Google Scholar 

  34. P. Erdős, A. Rényi, On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)

    Google Scholar 

  35. P. Erdős, A. Rényi, On the evolution of random graphs. Bull. Inst. Int. Statist. Tokyo 38, 343–347 (1961)

    Google Scholar 

  36. R. Fagin, Probabilities in finite models. J. Symbolic Logic 41, 50–58 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  37. K.J. Falconer, The realization of distances in measurable subsets covering \({\mathbb{R}}^{n}\). J. Combin. Theor. A 31, 187–189 (1981)

    Article  Google Scholar 

  38. F.M. de Oliveira Filho, F. Vallentin, Fourier analysis, linear programming, and densities of distance avoiding sets in \({\mathbb{R}}^{n}\). J. Eur. Math. Soc. 12, 1417–1428 (2010)

    Article  MATH  Google Scholar 

  39. V.P. Filimonov, Covering planar sets. Sb. Math. 201(N8), 1217–1248 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. V.P. Filimonov, Division codes and an area theorem. Dokl. Math. 79(N3), 345–348 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. V.P. Filimonov, Small deviation theorem for division codes. Dokl. Math. 81(N2), 278–281 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. P. Frankl, V. Rödl, Forbidden intersections. Trans. Am. Math. Soc. 300(N1), 259–286 (1987)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Yu.V. Glebskii, D.I. Kogan, M.I. Legon’kii, V.A. Talanov, Range and degree of realizability of formulas in the restricted predicate calculus. Cybernetics 5, 142–154 (1969)

    Google Scholar 

  45. E.S. Gorskaya, I.M. Mitricheva, V.Y. Protasov, A.M. Raigorodskii, Estimating the chromatic numbers of Euclidean spaces by methods of convex minimization. Sb. Math. 200(N6), 783–801 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. P.M. Gruber, C.G. Lekkerkerker, Geometry of Numbers (North-Holland, Amsterdam, 1987)

    MATH  Google Scholar 

  47. A.E. Guterman, V.K. Lyubimov, A.M. Raigorodskii, A.S. Usachev, On the independence numbers of distance graphs with vertices at { − 1, 0, 1}n: estimates, conjectures, and applications to the Borsuk and Nelson–Erdős–Hadwiger problems. J. Math. Sci. 165(N6), 689–709 (2010)

    Article  MathSciNet  Google Scholar 

  48. H. Hadwiger, Ein Überdeckungssatz für den Euklidischen Raum. Portugaliae Math. 4, 140–144 (1944)

    MathSciNet  MATH  Google Scholar 

  49. H. Hadwiger, Überdeckung einer Menge durch Mengen kleineren Durchmessers. Comm. Math. Helv. 18, 73–75 (1945/46); Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers. Comm. Math. Helv. 19, 72–73 (1946/47)

    Google Scholar 

  50. A. Heppes, P. Révész, Zum Borsukschen Zerteilungsproblem. Acta Math. Acad. Sci. Hung. 7, 159–162 (1956)

    Article  MATH  Google Scholar 

  51. L.L. Ivanov, On the chromatic numbers of \({\mathbb{R}}^{2}\) and \({\mathbb{R}}^{3}\) with intervals of forbidden distances. Vestnik of the Russian University of People’s Friendship (N1), 14–29 (2011) [in Russian]

    Google Scholar 

  52. L.L. Ivanov, An estimate of the chromatic number of the space \({\mathbb{R}}^{4}\). Russ. Math. Surv. 61(N5), 984–986 (2006)

    Article  MATH  Google Scholar 

  53. P.D. Johnson, Coloring abelian groups. Discrete Math. 40, 219–223 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  54. P.D. Johnson, Coloring the rational points to forbid the distance 1—a tentative history and compendium. Geombinatorics 16, 209–218 (2006)

    MathSciNet  Google Scholar 

  55. H.W.E. Jung, Über die kleinste Kugel, die eine räumliche Figur einschliesst. J. Reine und Angew. Math. 123, 241–257 (1901)

    MATH  Google Scholar 

  56. J. Kahn, G. Kalai, A counterexample to Borsuk’s conjecture. Bull. (new series) AMS 29(N1), 60–62 (1993)

    Google Scholar 

  57. J.-H. Kang, Z. Füredi, Distance graphs on \({\mathbb{Z}}^{n}\) with l 1-norm. Theor. Comp. Sci. 319(N1–3), 357–366 (2004)

    MATH  Google Scholar 

  58. Y. Katznelson, Chromatic numbers of Cayley graphs on Z and recurrence. Combinatorica 21, 211–219 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  59. V. Klee, S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory (Mathematical Association of America, Washington, DC, 1991)

    MATH  Google Scholar 

  60. R. Knast, An approximative theorem for Borsuk’s conjecture. Proc. Cambridge Phil. Soc. (N1), 75–76 (1974)

    Article  MathSciNet  Google Scholar 

  61. A.B. Kupavskiy, On coloring spheres embedded into \({\mathbb{R}}^{n}\). Sb. Math. 202(N6), 83–110 (2011)

    Article  MathSciNet  Google Scholar 

  62. A.B. Kupavskiy, On lifting of estimation of chromatic number of \({\mathbb{R}}^{n}\) in higher dimension. Dokl. Math. 429(N3), 305–308 (2009)

    Google Scholar 

  63. A.B. Kupavskii, On the chromatic number of \({\mathbb{R}}^{n}\) with an arbitrary norm. Discrete Math. 311, 437–440 (2011)

    Article  MathSciNet  Google Scholar 

  64. A.B. Kupavskii, The chromatic number of \({\mathbb{R}}^{n}\) with a set of forbidden distances. Dokl. Math. 82(N3), 963–966 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  65. A.B. Kupavskiy, A.M. Raigorodskii, On the chromatic number of \({\mathbb{R}}^{9}\). J. Math. Sci. 163(N6), 720–731 (2009)

    Article  MathSciNet  Google Scholar 

  66. A.B. Kupavskii, A.M. Raigorodskii, Counterexamples to Borsuk’s conjecture on spheres of small radii, submitted to Moscow J. Comb. and Number Theory. It is accepted and will be published in 2(3) (2012). available at arXiv: 1010.0383

    Google Scholar 

  67. D.G. Larman, A note on the realization of distances within sets in Euclidean space. Comment. Math. Helvet. 53, 529–535 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  69. M. Lassak, An estimate concerning Borsuk’s partition problem. Bull. Acad. Polon. Sci. Ser. Math. 30, 449–451 (1982)

    MathSciNet  MATH  Google Scholar 

  70. H. Lenz, Zerlegung ebener Bereiche in konvexe Zellen von möglichst kleinern Durchmesser. Jahresbericht d. DMV Bd. 58, 87–97 (1956)

    MathSciNet  MATH  Google Scholar 

  71. H. Lenz, Über die Bedeckung ebener Punktmengen durch solche kleineren Durchmessers. Arch. Math. VII, 34–40 (1956)

    Google Scholar 

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

    MATH  Google Scholar 

  73. V.K. Lyubimov, A.M. Raigorodskii, On lower bounds for the independence numbers of some distance graphs with vertices at { − 1, 0, 1}n. Dokl. Math. 80(N1), 547–549 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  74. F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977)

    MATH  Google Scholar 

  75. M. Mann, Hunting unit-distance graphs in rational n-spaces. Geombinatorics 13(N2), 49–53 (2003)

    Google Scholar 

  76. J. Matou\(\mathrm{\check{s}}\)ek, Using the Borsuk–Ulam Theorem, Universitext (Springer, Berlin, 2003)

    Google Scholar 

  77. N.G. Moshchevitin, A.M. Raigorodskii, On colouring the space \({\mathbb{R}}^{n}\) with several forbidden distances. Math. Notes 81(N5), 656–664 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  78. V.F. Moskva, A.M. Raigorodskii, New lower bounds for the independence numbers of distance graphs with vertices at { − 1, 0, 1}n. Math. Notes 89(N2), 307–308 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  79. O. Nechushtan, Note on the space chromatic number. Discrete Math. 256, 499–507 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  80. Y. Peres, W. Schlag, Two Erdős problems on lacunary sequences: chromatic numbers and Diophantine approximations, Bull. London Math. Soc. 42(2), 295–300 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  81. R. Radoi\(\mathrm{\check{c}}\)ić, G. Tóth, Note on the Chromatic Number of the Space. Discrete and Computational Geometry: The Goodman–Pollack Festschrift, 695–698. (Algorithms and Combinatorics, 25) (Springer, Berlin, 2003)

    Google Scholar 

  82. A.M. Raigorodskii, Borsuk’s problem and the chromatic numbers of metric spaces. Russ. Math. Surv. 56(N1), 103–139 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  83. A.M. Raigorodskii, The Chromatic Numbers (Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, 2003) [in Russian]

    Google Scholar 

  84. A.M. Raigorodskii, Borsuk’s Problem (Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, 2006) [in Russian]

    Google Scholar 

  85. A.M. Raigorodskii, Around Borsuk’s conjecture. J. Math. Sci. 154(N4), 604–623 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  86. A.M. Raigorodskii, Three lectures on the Borsuk partition problem. Lond. Math. Soc. Lect. Note Ser. 347, 202–248 (2007)

    MathSciNet  Google Scholar 

  87. A.M. Raigorodskii, The Borsuk partition problem: the seventieth anniversary. Math. Intel. 26(N3), 4–12 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  88. A.M. Raigorodskii, On the chromatic number of a space. Russ. Math. Surv. 55(N2), 351–352 (2000)

    Article  MathSciNet  Google Scholar 

  89. A.M. Raigorodskii, The Linear Algebra Method in Combinatorics (Moscow Centre for Continuous Mathematical Education (MCCME), Moscow, 2007) [in Russian]

    Google Scholar 

  90. A.M. Raigorodskii, On the chromatic number of a space with l q -norm. Russ. Math. Surv. 59(N5), 973–975 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  91. A.M. Raigorodskii, On the chromatic numbers of spheres in Euclidean spaces. Dokl. Math. 81(N3), 379–382 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  92. A.M. Raigorodskii, On the chromatic numbers of spheres in \({\mathbb{R}}^{n}\), Combinatorica, 32 (N1), 111–123 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  93. A.M. Raigorodskii, On distance graphs with large chromatic number but not containing large simplexes. Russ. Math. Surv. 62(N6), 1224–1225 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  94. A.M. Raigorodskii, The Erdős–Hadwiger problem and the chromatic numbers of finite geometric graphs. Sb. Math. 196(N1), 115–146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  95. A.M. Raigorodskii, On a bound in Borsuk’s problem. Russ. Math. Surv. 54(N2), 453–454 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  96. A.M. Raigorodskii, Counterexamples to Borsuk’s conjecture on spheres of small radius. Dokl. Math. 82(N2), 719–721 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  97. A.M. Raigorodskii, On lower bounds for the Borsuk and Hadwiger numbers. Russ. Math. Surv. 59(N3), 585–586 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  98. A.M. Raigorodskii, On the Borsuk and Erdős – Hadwiger numbers. Math. Notes 79(N6), 854–863 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  99. A.M. Raigorodskii, M.I. Absalyamova, A lower bound for the chromatic number of the space \({\mathbb{R}}^{n}\) with k forbidden distances and metric l 1. Chebyshev Sbornik 7(N4 (20)), 105–113 (2006) [in Russian]

    Google Scholar 

  100. A.M. Raigorodskii, M.M. Kityaev, On a series of questions concerning the Borsuk and the Nelson–Erdős–Hadwiger problems. Math. Notes 84(N2), 239–255 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  101. A.M. Raigorodskii, A.B. Kupavskiy, Partition of three-dimensional sets into five parts of smaller diameter. Math. Notes 87(N2), 208–219 (2010)

    Google Scholar 

  102. A.M. Raigorodskii, A.B. Kupavskii, M.V. Titova, On the densest sets without distance 1in spaces of small dimensions, Trudy of the Moscow Institute of Physics and Technology, 4, (N1), pp. 111–121 (2012)

    Google Scholar 

  103. A.M. Raigorodskii, O.I. Rubanov, in On the Clique and the Chromatic Numbers of High-Dimensional Distance Graphs, ed. by S.D. Adhikari, B. Ramakrishnan, Harish-Chandra Research Institute. Number Theory and Applications: Proceedings of the International Conferences on Number Theory and Cryptography (A publication of It’s located at Gurgaon, India, 2009), pp. 149–157

    Google Scholar 

  104. A.M. Raigorodskii, O.I. Rubanov, On distance graphs with large chromatic number and without large cliques. Math. Notes 87(N3), 417–428 (2010)

    MathSciNet  Google Scholar 

  105. A.M. Raigorodskii, I.M. Shitova, On the chromatic numbers of real and rational spaces with several real or rational forbidden distances. Sb. Math. 199(N4), 579–612 (2008)

    Article  MathSciNet  Google Scholar 

  106. A.M. Raigorodskii, I.M. Shitova, On the chromatic number of the Euclidean space and on Borsuk’s problem. Math. Notes 83(N4), 579–582 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  107. C. Richter, A. Hinrichs, New sets with large Borsuk numbers. Discrete Math. 270, 136–146 (2003)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  109. C.A. Rogers, Symmetrical sets of constant width and their partitions. Mathematika 18, 105–111 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  110. O.I. Rubanov, Chromatic numbers of 3-dimensional distance graphs containing no tetrahedra. Math. Notes 82(N5–6), 718–721 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  111. O. Schramm, Illuminating sets of constant width. Mathematika 35, 180–189 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  114. A. Soifer, The Mathematical Coloring Book (Springer, Berlin, 2009)

    MATH  Google Scholar 

  115. L.A. Székely, Erdős on unit distances and the Szemerédi–Trotter theorems, Paul Erdős and his mathematics, bolyai series Budapest. J. Bolyai Math. Soc. 11, 649–666 (2002)

    Google Scholar 

  116. L.A. Székely, N.C. Wormald, Bounds on the measurable chromatic number of \({\mathbb{R}}^{n}\). Discrete Math. 75, 343–372 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  117. D.R. Woodall, Distances realized by sets covering the plane. J. Combin. Theor. A 14, 187–200 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  118. N. Wormald, A 4-chromatic graph with a special plane drawing. Aust. Mat. Soc. (Ser. A) 28, 1–8 (1979)

    Google Scholar 

  119. A.R. Yarmukhametov, On the connectivity of random distance graphs of a special type. Chebyshev Sbornik 10(N1 (29)), 95–108 (2009)

    Google Scholar 

  120. M.E. Zhukovskii, The weak zero-one laws for the random distance graphs. Dokl. Math. 81(N1), 51–54 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  121. M.E. Zhukovskii, The weak zero-one law for the random distance graphs. Theor. Prob. Appl. 55(N2), 344–350 (2010)

    MathSciNet  Google Scholar 

  122. M.E. Zhukovskii, The weak zero-one law for the random distance graphs. Vestnik Russ. Univ. People’s Friendship 2(N1), 11–25 (2010) [in Russian]

    Google Scholar 

  123. G.M. Ziegler, Coloring Hamming graphs, optimal binary codes, and the 0/1-Borsuk problem in low dimensions. Lect. Notes Comput. Sci. 2122, 159–171 (2001)

    Article  Google Scholar 

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Acknowledgements

This work is done under the financial support of the following grants: Grant 12-01-00683 of Russian Foundation for Basic Research, Grant MD-8390.2010.1 of the Russian President, Grant NSh-2519.2012.1 supporting leading scientific schools in Russia.

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Authors and Affiliations

  1. Mechanics and Mathematics Faculty, Department of Mathematical Statistics and Random Processes, Moscow State University, GSP-1, 1 Leninskiye Gory, Main Building, 119991, Moscow, Russia

    Andrei M. Raigorodskii

  2. Faculty of Innovations and High Technology, Department of Data Analysis, Moscow Institute of Physics and Technology, Moscow, Russia

    Andrei M. Raigorodskii

  3. Yandex Research Laboratories, Moscow, Russia

    Andrei M. Raigorodskii

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  1. Andrei M. Raigorodskii
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  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

  2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

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Raigorodskii, A.M. (2013). Coloring Distance Graphs and Graphs of Diameters. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_23

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