Skip to main content
SpringerLink
Log in

The Hadwiger–Nelson Problem

  • Chapter
  • First Online:
Open Problems in Mathematics

Abstract

Inspired by the Four-Color Conjecture, the Hadwiger–Nelson Problem became one of the famous open problems of mathematics in its own rights. It has withstood all assaults for 65 years, and attracted many mathematicians from many fields, including Paul Erdős and Ronald L. Graham. John F. Nash admired this problem and chose it for the present book. In this chapter we will discuss this problem, its history and generalizations, several of the many related open problems, and the state of the art results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Unlike the previous version of this survey [55], more history of the problem and its authorship has been included here.

References

  1. H. Ardal, J. Manuch, M. Rosenfeld, S. Shelah and L. Stacho, The odd-distance plane graph, Discrete Comput. Geom. 42 (2009), 132–141.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Benda and M. Perles, Colorings of metric spaces, Geombinatorics IX (3) (2000), 113–126.

    Google Scholar 

  3. N. Brown, N. Dunfield and G. Perry, Colorings of the plane I, Geombinatorics III (2) (1993), 24–31.

    Google Scholar 

  4. N. Brown, N. Dunfield and G. Perry, Colorings of the plane II, Geombinatorics III (3) (1993), 64–74.

    Google Scholar 

  5. N. Brown, N. Dunfield and G. Perry, Colorings of the plane III, Geombinatorics III (4) (1993), 110–114.

    Google Scholar 

  6. K. Cantwell, All regular polytopes are Ramsey, J. Combin. Theory (A) 114 (2007), 555–562.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Cibulka, On the chromatic number of real and rational spaces, Geombinatorics XVIII (2) (2008), 53–65.

    Google Scholar 

  8. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd edn., Springer-Verlag, 1999.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Coulson, Tilings and colorings of 3-space, Geombinatorics XII (3) (2003), 102–116.

    Google Scholar 

  11. N. G. de Bruijn and P. Erdős, A color problem for infinite graphs and a problem in the theory of relations, Indag. Math. 13 (1951), 369–373.

    Google Scholar 

  12. J.-P. Delahaye, Coloriages irréals, Pour la Science, Février 2005, 88-93. English translation: Imaginary Coloring, Geombinatorics XV(3), 2006, 101-119.

    Google Scholar 

  13. P. Erdős, Problem (p. 681); Unsolved problems, Proceedings of the Fifth British Combinatorial Conference 1975, University of Aberdeen, July 14–18, 1975, (ed. C. St.J. A. Nash-Williams and J. Sheehan), Congr. Numer. XV, Utilitas Mathematica, 1976.

    Google Scholar 

  14. P. Erdős, Combinatorial problems in geometry and number theory, Relations between combinatorics and other parts of mathematics, Proc. Sympos. Pure Math., Ohio State Univ., (1978); Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc. (1979), 149–162.

    Google Scholar 

  15. P. Erdős, Some combinatorial problems in geometry, Geom. & Diff. Geom. Proc., Haifa, Israel, (1979); Lecture Notes in Math. 792, Springer (1980), 46–53.

    Google Scholar 

  16. P. Erdős, 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 

  17. P. Erdős, Video recording of the talk "Twenty Five Years of Questions and Answers", 25 th South-Eastern International Conference On Combinatorics, Graph Theory and Computing, Florida Atlantic University, Boca Raton, March 10, 1994.

    Google Scholar 

  18. K. J. Falconer, The realization of distances in measurable subsets covering Rn, J. Combin. Theory (A) 31 (1981), 187–189.

    Article  MathSciNet  MATH  Google Scholar 

  19. K. J. Falconer, and J. M. Marstrand, Plane sets with positive density at infinity contain all large distances. Bull. London Math. Soc. 18 (1986), 471–474.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Frankl and R. M. Wilson, Intersection theorems with geometrical consequences, Combinatorica 1 (1981), 357–368.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. L. Graham, Some of My Favourite Problems in Ramsey Theory, B. Landman et al. (ed’s), Proceedings of the ‘Integers Conference 2005’ in Celebration of the 70th Birthday of Ronald Graham, Carrolton, Georgia, USA, 2005, Walter de Gruyter (2007), 229–236.

    Google Scholar 

  22. R. L. Graham, Old and new problems and results in Ramsey theory, Horizons of Combinatorics (Conference and EMS Summer School), Budapest and Lake Balaton, Hungary, 2006, (Bolyai Society Mathematical Studies) (ed. E. Gyori, G. Katona and L. Lovasz), (2008), 105–118.

    Google Scholar 

  23. R. L. Graham, B. L. Rothschild, and E. G. Straus, Are there n + 2 points in E n with odd integral distances?, Amer. Math. Monthly 81 (1974) 21–25.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Hochberg and P. O’Donnell, Some 4-chromatic unit-distance graphs without small cycles, Geombinatorics V (4) (1996), 137–141.

    Google Scholar 

  25. I. Hoffman and A. Soifer, Almost chromatic number of the plane, Geombinatorics III (2) (1993), 38–40.

    Google Scholar 

  26. I. Hoffman and A. Soifer, Another six-coloring of the plane, Discrete Math. 150 (1996), 427–429.

    Article  MathSciNet  MATH  Google Scholar 

  27. P. D. Johnson, Jr., Coloring the rational points to forbid the distance one – A tentative history and compendium, Geombinatorics XVI (1) (2006), 209–218.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Mann, A new bound for the chromatic number of the rational five-space, Geombinatorics XI (2) (2001), 49–53.

    Google Scholar 

  30. M. Mann, Hunting unit-distance graphs in rational n-spaces, Geombinatorics XIII (2) (2003), 86–97.

    Google Scholar 

  31. L. Moser and W. Moser, Solution to problem 10, Canad. Math. Bull. 4 (1961), 187–189.

    MATH  Google Scholar 

  32. O. Nechushtan, On the space chromatic number, Discrete Math. 256 (2002), 499–507.

    Article  MathSciNet  MATH  Google Scholar 

  33. E. Nelson, Letter to A. Soifer of October 5, 1991.

    Google Scholar 

  34. P. O’Donnell, High Girth Unit-distance Graphs, Ph.D. thesis, Rutgers University, 1999.

    Google Scholar 

  35. M. S. Payne, A unit distance graph with ambiguous chromatic number, arXiv: 0707.1177v1 [math.CO], 9 July 2007.

    Google Scholar 

  36. M. S. Payne, Unit distance graphs with ambiguous chromatic number, Electronic J. Combin. 16 (2009).

    Google Scholar 

  37. Materialy Konferenzii “Poisk-97”, Moscow, 1997 (Russian).

    Google Scholar 

  38. D. Pritikin, All unit-distance graphs of order 6197 are 6-colorable, J. Combin. Theory (B) (1998), 159–163.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. D. E. Raiskii, Realizing of all distances in a decomposition of the space R n into n + 1 parts, Mat. Zametki 7 (1970), 319–323 [Russian]; English transl., Math. Notes 7 (1970), 194–196.

    Google Scholar 

  41. M. Rosenfeld, Odd integral distances among points in the plane, Geombinatorics V (4) (1996), 156–159.

    Google Scholar 

  42. S. Shelah and A. Soifer, Axiom of choice and chromatic number of the plane, J. Combin. Theory (A) 103 (2003), 387–391.

    Google Scholar 

  43. A. Soifer, Relatives of chromatic number of the plane I, Geombinatorics I (4) (1992), 13–15.

    Google Scholar 

  44. A. Soifer, A six-coloring of the plane, J. Comb. Theory (A) 61 (1992), 292–294.

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Soifer, Six-realizable set X6, Geombinatorics III (4), 1994, 140-145.

    Google Scholar 

  46. A. Soifer, An infinite class of 6-colorings of the plane, Congr. Numer. 101 (1994), 83–86.

    MathSciNet  MATH  Google Scholar 

  47. A. Soifer, Axiom of choice and chromatic number of R n, J. Combin. Theory (A), 110 (2005), 169–173.

    Google Scholar 

  48. A. Soifer and S. Shelah, Axiom of choice and chromatic number: an example on the plane, J. Combin. Theory (A) 105 (2004), 359–364.

    Google Scholar 

  49. A. Soifer, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, Springer, New York, 2009.

    MATH  Google Scholar 

  50. R. M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1–56.

    Article  MathSciNet  MATH  Google Scholar 

  51. C. Thomassen, On the Nelson unit distance coloring problem, Amer. Math. Monthly, 106 (1999), 850–853

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  53. N. C. Wormald, A 4-chromatic graph with a special plane drawing, J. Austral. Math. Soc. (A) 28 (1979), 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  54. V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, Math. Assoc. Amer., 2nd edn (with Addendum), Washington D.C., 1991.

    Google Scholar 

  55. A. Soifer, Geometric graphs, in L.W. Beineke and R.J. Wilson (ed's) Topics in Chromatic Number Theory, Cambridge University Press, 2015, 161–180.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Colorado, 1420 Austin Bluffs Parkway, Colorado Springs, CO, 80918, USA

    Alexander Soifer

Authors
  1. Alexander Soifer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Soifer .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Senior Research Mathematician, Princeton University, Princeton, New Jersey, USA

    John Forbes Nash, Jr.

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Michael Th. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Soifer, A. (2016). The Hadwiger–Nelson Problem. In: Nash, Jr., J., Rassias, M. (eds) Open Problems in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32162-2_14

Download citation

Publish with us

Policies and ethics

Search

Navigation