Advertisement

Ramsey Theory pp 115-120 | Cite as

Open Problems in Euclidean Ramsey Theory

  • Ron Graham
  • Eric Tressler
Chapter
Part of the Progress in Mathematics book series (PM, volume 285)

Abstract

Ramsey theory is the study of structure that must exist in a system, most typically after it has been partitioned.

Keywords

Hexagonal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    H. Ardal, J. Maňuch, M. Rosenfeld, S. Shelah, and L. Stacho, The odd-distance plane graph. Discrete Comput. Geom., 42 (2009), 132–141.Google Scholar
  2. [2]
    M. Benda and M. Perles, Colorings of metric spaces. Geombinatorics, 9 (2000), 113–126.Google Scholar
  3. [3]
    K. B. Chilakamarri, Some problems arising from unit–distance graphs. Geombinatorics, 4(4) (1995), 104–109.Google Scholar
  4. [4]
    K. B. Chilakamarri, On the chromatic number of rational five-space. Aequationes Mathematicae, 39 (1990), 146–148.Google Scholar
  5. [5]
    D. A. Coulson, A 15-coloring of 3-space omitting distance one, Discrete Math. 256 (2002), 83–90.Google Scholar
  6. [6]
    D. A. Coulson and M. S. Payne, A dense distance 1 excluding set in 3. Aust. Math. Soc. Gazette, 34 (2007), 97–102.Google Scholar
  7. [7]
    H. Croft, Incidence incidents. Eureka, 30 (1967), 22–26.Google Scholar
  8. [8]
    P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus, Euclidean Ramsey Theory I, J. Combin. Theor. Ser. A 14 (1973), 341–363.Google Scholar
  9. [9]
    P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. G. Straus, Euclidean Ramsey Theory III, Infinite and finite sets Colloq., Keszthely (1973), Vol. I, pp. 559–583. Colloq. Math. Soc. Janos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975.Google Scholar
  10. [10]
    P. Frankl and V. Rödl, A partition property of simplices in Euclidean space. J. Am. Math. Soc., 3 (1990), 1–7.Google Scholar
  11. [11]
    L. L. Ivanov, An estimate for the chromatic number of the space 4. Russ. Math. Surv., 61 (2006), 984–986.Google Scholar
  12. [12]
    L. L. Ivanov, On the chromatic numbers of 2 and 3 with intervals of forbidden distances. Electron. Notes Discrete Math., 29 (2007), 159–162.Google Scholar
  13. [13]
    V. Jelínek, J. Kynčl, R. Stolař, and T. Valla, Monochromatic triangles in two-colored plane. arXiv.math/0701940v1.Google Scholar
  14. [14]
    I. Křiž, Permutation groups in Euclidean Ramsey theory. Proc. Am. Math. Soc. 112 (1991), 899–907.Google Scholar
  15. [15]
    I. Křiž, All trapezoids are Ramsey. Discrete Math. 108 (1992), 59–62.Google Scholar
  16. [16]
    K. Kuratowski, Sur le problème des courbes gauches en topologie. Fund. Math. 15 (1930), 271–283.Google Scholar
  17. [17]
    O. Nechushtan, On the space chromatic number. Discrete Math., 256 (2002), 499–507.Google Scholar
  18. [18]
    P. O’Donnell, Arbitrary girth, 4-chromatic unit distance graphs in the plane. I. Graph description. Geombinatorics 9 (2000), 145–152.Google Scholar
  19. [19]
    P. O’Donnell, Arbitrary girth, 4-chromatic unit distance graphs in the plane. II. Graph embedding. Geombinatorics 9 (2000), 180–193.Google Scholar
  20. [20]
    A. M. Raǐgorodoskiǐ and I. M. Shitova, Chromatic numbers of real and rational spaces with real or rational forbidden distances. Sbornik: Math., 199(4), 579–612.Google Scholar
  21. [21]
    D. E. Raiskii, Realizing of all distances in a decomposition of the space n into n+1 parts. Zametki, 9 (1970), 319–323.Google Scholar
  22. [22]
    L. Shader, All right triangles are Ramsey in E 2! J. Comb. Theor., Ser. A, 20 (1976), 385–389.Google Scholar
  23. [23]
    A. Soifer, Chromatic number of the plane & its relatives. I. The problem & its history. Geombinatorics 12 (2003), 131–148.Google Scholar
  24. [24]
    A. Soifer, Chromatic number of the plane & its relatives. II. Polychromatic number & 6-coloring. Geombinatorics 12 (2003), 191–216.Google Scholar
  25. [25]
    A. Soifer, Chromatic number of the plane & its relatives. III. Its future. Geombinatorics 13 (2003), 41–46.Google Scholar
  26. [26]
    A. Soifer, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators. Springer, New York, 2009.MATHGoogle Scholar
  27. [27]
    E. G. Straus Jr., A combinatorial theorem in group theory, Math. Comp. 29 (1975), 303–309.Google Scholar
  28. [28]
    B. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskunde 19 (1927), 212–216.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Ron Graham
    • 1
  • Eric Tressler
    • 2
  1. 1.Department of Computer ScienceUniversity of CaliforniaSan DiegoUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

Personalised recommendations