Ramsey Theory pp 121-161 | Cite as

Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts

  • Alexander Soifer
Part of the Progress in Mathematics book series (PM, volume 285)


In August 1987 I attended an inspiring talk by Paul Halmos at Chapman College in Orange, California. It was entitled “Some Problems You Can Solve, and Some You Cannot.” This problem was an example of a problem “you cannot solve.”


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  1. [BE2]
    Bruijn, N. G. de, and Erdős, P., A colour problem for infinite graphs and a problem in the theory of relations, Indagationes Math. 13 (1951), 369–373.Google Scholar
  2. [BP]
    Benda, M., and Perles, M., Colorings of metric spaces, Geombinatorics IX(3), (2000), 113–126.Google Scholar
  3. [BMP]
    Brass, P., Moser, W., and Pach, J., Research Problems in Discrete Geometry, Springer, New York, 2006.Google Scholar
  4. [BDP1]
    Brown, N., Dunfield, N., and Perry, G., Colorings of the plane I, Geombinatorics III(2), (1993), 24–31.Google Scholar
  5. [BDP2]
    Brown, N., Dunfield, N., and Perry, G., Colorings of the plane II, Geombinatorics III(3), (1993), 64–74.Google Scholar
  6. [BDP3]
    Brown, N., Dunfield, N., and Perry, G., Colorings of the plane III, Geombinatorics III(4), (1993), 110–114.Google Scholar
  7. [Can1]
    Cantwell, K., All regular polytopes are Ramsey, J. Combin. Theory Ser A 114 (2007) 555–562.MathSciNetMATHCrossRefGoogle Scholar
  8. [Chi1]
    Chilakamarri, K. B., Unit-distance graphs in rational n-spaces, Discrete Math. 69 (1988), 213–218.MathSciNetMATHCrossRefGoogle Scholar
  9. [Chi2]
    Chilakamarri, K. B., On the chromatic number of rational five-space, Aequationes Math. 39 (1990), 146–148.MathSciNetMATHCrossRefGoogle Scholar
  10. [Chi4]
    Chilakamarri, K. B., The unit-distance graph problem: A brief survey and some new results, Bull. ICA 8 (1993), 39–60.MathSciNetMATHGoogle Scholar
  11. [Cho]
    Chow, T., Distances forbidden by two-colorings of Q 3 and A n (manuscript).Google Scholar
  12. [Cib]
    Cibulka, J., On the chromatic number of real and rational spaces, Geombinatorics XVIII(2), (2008), 53–65.Google Scholar
  13. [CS]
    Conway, J. H., and Sloane, N. J. A. Sphere Packings, Lattices and Groups, 3rd ed., with contrib. by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen, and B. B. Venkov, Springer, New York, 1999.Google Scholar
  14. [Cou2]
    Coulson, D., A 15-colouring of 3-space omitting distance one, Discrete Math. 256 (1–2), (2002), 83–90.Google Scholar
  15. [Cou3]
    Coulson, D., Tilings and colourings of 3-space, Geombinatorics XII (3), (2003), 102–116.Google Scholar
  16. [Cro]
    Croft, H. T., Incidence incidents, Eureka (Cambridge) 30 (1967), 22–26.Google Scholar
  17. [CFG]
    Croft, H. T., Falconer, K. J., and Guy, R. K., Unsolved Problems in Geometry, Springer, New York, 1991.MATHCrossRefGoogle Scholar
  18. [DeM4]
    De Morgan, A., Review of the philosophy of discovery…, The Athenaeum 1694, London, 1860, 501–503.Google Scholar
  19. [E61.05]
    Erdős, P., Graph theory and probability II, Can. J. Math. 13 (1961), 346–352.CrossRefGoogle Scholar
  20. [E61.22]
    Erdős, P., Some unsolved problems, Magyar tudomanyos academia mathematikai kutato intezetenek közlemenyel 6 ser. A(1–2), (1961), 221–254.Google Scholar
  21. [E63.21]
    Erdős, P., Nekotorye Nereshennye Problemy (Some unsolved problems), Matematika 7(4), (1963), 109–143. (A Russian translation of [E61.22].)Google Scholar
  22. [E75.24]
    Erdős, P., On some problems of elementary and combinatorial geometry, Annali Matematica Pura Applicata 103(4), (1975), 99–108.Google Scholar
  23. [E76.49]
    Erdős, P., Problem (p. 681); P. Erdős, Chairman, Unsolved problems, in Proceedings of the Fifth British Combinatorial Conference 1975, University of Aberdeen, July 14–18, 1975, C. St. J. A. Nash-Williams and J. Sheehan, eds., Congressus Numerantium XV, Winnipeg, Utilitas Mathematica, 1976.Google Scholar
  24. [E78.50]
    Erdős, P., Problem 28 in “Exercises, problems and conjectures,” in Bollobás, B., Extremal Graph Theory, London Mathematical Society Monographs, Academic Press, 1978, p. 285.Google Scholar
  25. [E79.04]
    Erdős, P., Combinatorial problems in geometry and number theory, Relations between combinatorics and other parts of mathematics, Proc. of the Sympos. Pure Math., Ohio State Univ., Columbus, OH, (1978); Proc. of the Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, RI, (1979), 149–162.Google Scholar
  26. [E80.38]
    Erdős, P., Some combinatorial problems in geometry, Geom. & Diff. Geom. Proc., Haifa, Israel, (1979); Lecture Notes in Math., 792, Springer (1980), 46–53.Google Scholar
  27. [E80.41]
    Erdős, P., Some combinational problems in geometry, Geometry and Differential Geometry (Proc. Conf., Univ. Haifa, Haifa, 1979), Lecture Notes in Math., 792, pp. 46–53, Springer, Berlin, 1980.Google Scholar
  28. [E81.23]
    Erdős, P., Some applications of graph theory and combinatorial methods to number theory and geometry, Algebraic Methods in Graph Theory (Colloq. held in Szeged, Hungary 1978), Vol. I, Coll. Math Soc. J. Bolyai 25 (1981), 137–148.Google Scholar
  29. [E81.26]
    Erdős, P., Some new problems and results in graph theory and other branches of combinatorial mathematics, Combinatorics and Graph Theory, Proc. Symp. Calcutta 1980, Lecture Notes Math., 885, Springer, 1981, 9–17.Google Scholar
  30. [E85.01]
    Erdős, P., Problems and results in combinatorial geometry, Discrete Geometry and Convexity, Annals of the New York Academy of Sciences, 440, The New York Academy of Sciences, New York, 1985, 1–11.Google Scholar
  31. [E91/7/16]
    Erdős, P., Letter to A. Soifer of July 16, 1991.Google Scholar
  32. [E91/8/10]
    Erdős, P., Letter to A. Soifer of August 10, 1991.Google Scholar
  33. [E91/10/2]
    Erdős, P., Letter to A. Soifer, received on October 2, 1991 (marked “1977 VII 25”).Google Scholar
  34. [E92.19]
    Erdős, P., On some unsolved problems in elementary geometry (Hungarian), Mat. Lapok (N.S.) 2(2), (1992), 1–10.Google Scholar
  35. [E92.60]
    Erdős, P., Video recording of the talk “Some of my favorite problems II”, University of Colorado at Colorado Springs, January 10, 1992.Google Scholar
  36. [E94.60]
    Erdős, P., Video recording of the talk “Twenty five years of questions and answers,” 25th South-Eastern International Conference On Combinatorics, Graph Theory and Computing, Florida Atlantic University, Boca Raton, March 10, 1994.Google Scholar
  37. [EHT]
    Erdős, P., Harary, F., and Tutte, W. T., On the dimension of a graph, Mathematica 12(1965), 118–125.Google Scholar
  38. [ESi]
    Erdős, P., and Simonovits, M., On the chromatic number of geometric graphs, Ars Comb. 9 (1980), 229–246.Google Scholar
  39. [Fal1]
    Falconer, K. J., The realization of distances in measurable subsets covering R n, J.Combin. Theory (A) 31 (1981), 187–189.Google Scholar
  40. [Fis1]
    Fischer, K. G., Additive k-colorable extensions of the rational plane, Discrete Math. 82 (1990), 181–195.MathSciNetMATHCrossRefGoogle Scholar
  41. [Fis2]
    Fischer, K. G., The connected components of the graph \(Q(Q{(\sqrt{{N}_{1}},\ldots,\sqrt{{N}_{d}})}^{2}\), Congressus Numerantium 72 (1990), 213–221.MathSciNetGoogle Scholar
  42. [FW]
    Frankl, P., and Wilson, R. M., Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357–368.MathSciNetMATHCrossRefGoogle Scholar
  43. [Gars2]
    Garsia, A., E-mail to A. Soifer, February 28, 1995.Google Scholar
  44. [Gra1]
    Graham, R. L., Talk at the Mathematical Sciences Research Institute, Berkeley, August 2003.Google Scholar
  45. [Gra2]
    Graham, R. L., Some of my favorite problems in Ramsey theory, Proceedings of the ‘Integers Conference 2005’ in Celebration of the 70th Birthday of Ronald Graham, Carrolton, Georgia, USA, October 27–30, 2005, B. Landman et al., ed., Walter de Gruyter, Berlin, 2007, 229–236.Google Scholar
  46. [Gra3]
    Graham, R. L., Old and new problems and results in Ramsey theory, Horizon of Combinatorics (Conference and EMS Summer School), Budapest and Lake Balaton, Hungary, 10–22 July 2006, to appear.Google Scholar
  47. [GT]
    Graham, R. L., and Tressler, E., Open problems in Euclidean Ramsey theory, in this volume.Google Scholar
  48. [GO]
    Goodman, J. E., and O’Rourke, J., Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, FL, 1997.MATHGoogle Scholar
  49. [Had3]
    Hadwiger, H., Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.MathSciNetMATHGoogle Scholar
  50. [Had4]
    Hadwiger, H., Ungelöste Probleme, Nr. 11, Elemente der Mathematik 16 (1961), 103–104.Google Scholar
  51. [HD1]
    Hadwiger, H., and Debrunner, H., Ausgewählte einzelprobleme der kombinatorishen geometrie in der ebene, L’Enseignement Mathematique 1 (1955), 56–89.MathSciNetMATHGoogle Scholar
  52. [HD2]
    Hadwiger, H., and Debrunner, H., Kombinatorischen Geometrie in der Ebene, L’Enseignement Mathematique, Geneva, 1959.Google Scholar
  53. [HD3]
    Hadwiger, H., and Debrunner, H., Combinatorial Geometry of the Plane ed. I. M. Yaglom, Nauka, Moscow, 1965 (extended by the editor Russian Translation version of [HD2]).Google Scholar
  54. [HDK]
    Hadwiger, H., Debrunner, H., and Klee, V., Combinatorial Geometry in the Plane, Holt, Rinehart and Winston, NY, 1964.Google Scholar
  55. [HO]
    Hochberg, R., and O’Donnell, P., Some 4-chromatic unit-distance graphs without small cycles, Geombinatorics V(4), (1996), 137–14.Google Scholar
  56. [HS1]
    Hoffman, I., and Soifer, A., Almost chromatic number of the plane, Geombinatorics III(2), (1993), 38–40.Google Scholar
  57. [HS2]
    Hoffman, I., and Soifer, A., Another six-coloring of the plane, Discrete Math. 150 (1996), 427–429.MathSciNetMATHCrossRefGoogle Scholar
  58. [Isb1]
    Isbell, J., Letter to A. Soifer of August 26, 1991.Google Scholar
  59. [Isb2]
    Isbell, J., Letter to A. Soifer of September 3, 1991.Google Scholar
  60. [Joh1]
    Johnson Jr., P. D., Coloring Abelian groups, Discrete Math. 40 (1982), 219–223.MathSciNetMATHCrossRefGoogle Scholar
  61. [Joh2]
    Johnson Jr., P. D., Simple product colorings, Discrete Math. 48 (1984), 83–85.MathSciNetMATHCrossRefGoogle Scholar
  62. [Joh3]
    Johnson Jr., P. D., Two-coloring of real quadratic extensions of Q 2 that forbid many distances, Congressus Numerantium 60 (1987), 51–58.MathSciNetGoogle Scholar
  63. [Joh4]
    Johnson Jr., P. D., Two-colorings of a dense subgroup of Q n that forbid many distances, Discrete Math. 79 (1989/1990), 191–195.Google Scholar
  64. [Joh5]
    Johnson Jr., P. D., Product colorings, Geombinatorics I(3), (1991), 11–12.Google Scholar
  65. [Joh6]
    Johnson Jr., P. D., Maximal sets of translations forbidable by two-colorings of Abelian groups, Congressus Numerantium 91 (1992), 153–158.MathSciNetGoogle Scholar
  66. [Joh7]
    Johnson Jr., P. D., About two definitions of the fractional chromatic number, Geombinatorics V(3), (1996), 11–12.Google Scholar
  67. [Joh8]
    Johnson Jr., P. D., Introduction to “colorings of metric spaces” by Benda and Perles, Geombinatorics IX(3), (2000), 110–112.Google Scholar
  68. [Joh9]
    Johnson Jr., P. D., Coloring the rational points to forbid the distance one – A tentative history and compendium, Geombinatorics XVI(1), (2006), 209–218.Google Scholar
  69. [Joh10]
    Johnson Jr., P. D., Euclidean distance graphs on the rational points, appears in this volume.Google Scholar
  70. [JRW]
    Jungreis, D. S., Reid, M., and Witte, D., Distances forbidden by some 2-coloring of Q 2, Discrete Math. 82 (1990), 53–56.MathSciNetMATHCrossRefGoogle Scholar
  71. [KW]
    Klee, V., and Wagon, S., Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America, second edition (with addendum), 1991.Google Scholar
  72. [LR]
    Larman, D. G., and Rogers, C. A., The realization of distances within sets in Euclidean space, Mathematika 19 (1972), 1–24.MathSciNetMATHCrossRefGoogle Scholar
  73. [LV]
    Lovász, L., and Vesztergombi, K., Geometric representations of graphs, in Paul Erdös and his Mathematics II, Halás, G. et al. (ed.), Springer, 2002, 471–498.Google Scholar
  74. [Man1]
    Mann, M., A new bound for the chromatic number of the rational five-space, Geombinatorics XI(2), (2001), 49–53.Google Scholar
  75. [Man2]
    Mann, M., Hunting unit-distance graphs in rational n-spaces, Geombinatorics XIII(2), (2003), 86–97.Google Scholar
  76. [MM]
    Moser, L., and Moser, W., Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.Google Scholar
  77. [Nec]
    Nechushtan, O., On the space chromatic number, Discrete Math. 256 (2002), 499–507.MathSciNetMATHCrossRefGoogle Scholar
  78. [Nel1]
    Nelson, E., Letter to A. Soifer of August 23, 1991.Google Scholar
  79. [Nel2]
    Nelson, E., Letter to A. Soifer of October 5, 1991.Google Scholar
  80. [Odo3]
    O’Donnell, P., High girth unit-distance graphs, Ph. D. Dissertation, Rutgers University, May 25, 1999.Google Scholar
  81. [Odo4]
    O’Donnell, P., Arbitrary girth, 4-chromatic unit-distance graphs in the plane part I: Graph description, Geombinatorics IX(3), (2000), 145–150.Google Scholar
  82. [Odo5]
    O’Donnell, P., Arbitrary girth, 4-chromatic unit-distance graphs in the plane part II: Graph embedding, Geombinatorics IX(4), (2000), 180–193.Google Scholar
  83. [Pay1]
    Payne, M. S., A unit distance graph with ambiguous chromatic number, arXiv: 0707.1177v1 [math.CO] 9 Jul 2007.Google Scholar
  84. [Pay2]
    Payne, M. S., Unit distance graphs with ambiguous chromatic number, Electronic J.Combin. 16(1), 2009 (Nov 7, 2009).Google Scholar
  85. [Pet]
    Peter, L. J., Peter’s Quotations: Ideas for Our Time, William Morrow, New York, 1977.Google Scholar
  86. [Poisk]
    Materialy Konferenzii “Poisk-97”, Moscow, 1997, (Russian).Google Scholar
  87. [Pri]
    Pritikin, D., All unit-distance graphs of order 6197 are 6-colorable, J. Combin. Theory Ser. B 73(2), (1998), 159–163.Google Scholar
  88. [RT]
    Radoicic, R., and Tóth, G., Note on the chromatic number of the space, Discrete and Computational Geometry: the Goodman-Pollack Festschrift, B. Aronov, S. Basu, J. Pach, and M. Sharir (eds.), Springer, Berlin 2003, 695–698.Google Scholar
  89. [Raig2]
    Raigorodskii, A. M., On the chromatic number of a space, Russ. Math. Surv. 55(2), (2000), 351–352.Google Scholar
  90. [Raig3]
    Raigorodskii, A. M., Borsuk’s problem and the chromatic numbers of some metric spaces, Russian Math. Surv. 56(1), (2001), 103–139.Google Scholar
  91. [Raig6]
    Raigorodskii, A. M., Chromatic Numbers (Russian), Moscow Center of Continuous Mathematical Education, Moscow, 2003.Google Scholar
  92. [Rai]
    Raiskii, D. E., Realizing of all distances in a decomposition of the space R n into n+1 Parts, Mat. Zametki 7 (1970), 319–323 [Russian]; Engl. transl., Math. Notes 7 (1970), 194–196.Google Scholar
  93. [SS1]
    Shelah, S., and Soifer, A., Axiom of choice and chromatic number of the Plane, J.Combin. Theory Ser. A 103(2003) 387–391.MathSciNetMATHCrossRefGoogle Scholar
  94. [SS2]
    Soifer, A., and Shelah, S., Axiom of choice and chromatic number: An example on the plane, J. Combin. Theory Ser. A 105(2), (2004), 359–364.Google Scholar
  95. [Sim]
    Simmons, G. J., The chromatic number of the sphere, J. Austral. Math. Soc. Ser. A 21 (1976), 473–480.MathSciNetMATHCrossRefGoogle Scholar
  96. [Soi]
    Soifer, A., The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators, Springer, New York, 2009.MATHGoogle Scholar
  97. [Soi3]
    Soifer, A., Triangles in a three-colored plane, Geombinatorics I(2) (1991), 11–12 and I(4) (1992), 21.Google Scholar
  98. [Soi4]
    Soifer, A., Chromatic number of the plane: A historical essay, Geombinatorics I(3), (1991), 13–15.Google Scholar
  99. [Soi5]
    Soifer, A., Relatives of chromatic number of the plane I, Geombinatorics I(4), (1992), 13–15.Google Scholar
  100. [Soi6]
    Soifer, A., A six-coloring of the plane, J. Combin. Theory Ser A 61(2), (1992), 292–294.Google Scholar
  101. [Soi7]
    Soifer, A., Six-realizable set X 6, Geombinatorics III(4), (1994).Google Scholar
  102. [Soi8]
    Soifer, A., An infinite class of 6-colorings of the plane, Congressus Numerantium 101 (1994), 83–86.MathSciNetMATHGoogle Scholar
  103. [Soi17]
    Soifer, A., 50th Anniversary of one problem: Chromatic number of the plane, Mathematics Competitions 16(1), (2003), 9–41.Google Scholar
  104. [Soi23]
    Soifer, A., Axiom of choice and chromatic number of R n, J. Combin. Theory Ser A 110(1), (2005), 169–173.Google Scholar
  105. [Sol1]
    Solovay, R. M., A model of set theory in which every set of reals is Lebesgue measurable, Ann. Math. 92 Ser. 2 (1970), 1–56.Google Scholar
  106. [Sze1]
    Székely, L. A., Remarks on the chromatic number of geometric graphs, in Graphs and Other Combinatorial Topics, Proceedings of the Third Czechoslovak Symposium on Graph Theory, Prague, August 24–27, 1982, M. Fiedler, ed., Teubner Verlaggesellschaft, Leipzig, 1983, 312–315.Google Scholar
  107. [Sze2]
    Székely, L. A., Measurable chromatic number of geometric graphs and sets without some distances in Euclidean space, Combinatorica 4 (1984) 213–218.MathSciNetMATHCrossRefGoogle Scholar
  108. [Sze3]
    Szekeres, Gy., A combinatorial problem in geometry: Reminiscences, in Paul Erdös, The Art of Counting, MIT Press, Cambridge, MA, 1973, xix–xxii.Google Scholar
  109. [Woo1]
    Woodall, D. R., Distances realized by sets covering the plane, J. Combin. Theory Ser. A 14 (1973), 187–200.MathSciNetMATHCrossRefGoogle Scholar
  110. [Wor]
    Wormald, N.C., A 4-chromatic graph with a special plane drawing, J. Austral. Math. Soc. Ser. A 28(1979), 1–8.MathSciNetMATHCrossRefGoogle Scholar
  111. [Zak1]
    Zaks, J., On the chromatic number of some rational spaces, Combin. Optim., Research Report Corr 88–26 (1988), 1–3.Google Scholar
  112. [Zak2]
    Zaks, J., On four-colourings of the rational four-space, Aequationes Math. 27 (1989) 259–266.MathSciNetCrossRefGoogle Scholar
  113. [Zak4]
    Zaks, J., On the chromatic number of some rational spaces, Ars Combinatorica 33 (1992), 253–256.MathSciNetMATHGoogle Scholar
  114. [Zak6]
    Zaks, J., Uniform distances in rational unit-distance graphs, Discrete Math. 109 (1–3), (1992) 307–311.Google Scholar
  115. [Zak7]
    Zaks, J., On odd integral distances rational graphs, Geombinatorics IX(2), (1999), 90–95.Google Scholar

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Authors and Affiliations

  1. 1.University of Colorado at Colorado SpringsColorado SpringsUSA

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