The fractional chromatic number of the plane


The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance is 1. This chromatic number is denoted χ(ℝ2). The problem was introduced in 1950, and shortly thereafter it was proved that 4≤χ(ℝ2)χ≤7. These bounds are both easy to prove, but after more than 60 years they are still the best known. In this paper, we investigate χ f (ℝ2), the fractional chromatic number of the plane. The previous best bounds (rounded to five decimal places) were 3.5556≤χ f (ℝ2)≤4.3599. Here we improve the lower bound to 76/21≈3.6190.

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


  1. [1]

    H. Croft: Incidence Incidents, Eureka 30 (1967), 22–26.

    Google Scholar 

  2. [2]

    N. G. de Bruijn and P. Erdős: A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951), 369–373.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    P. Erdős: Unsolved problems, Congressus Numerantium XV–Proceedings of the 5th British Combinatorial Conference. 1975, 681, 1976.

    Google Scholar 

  4. [4]

    P. Erdős: Problems and results in combinatorial geometry, In Discrete geometry and convexity (New York, 1982), volume 440 of Ann. New York Acad. Sci., pages 1–11. New York Acad. Sci., New York, 1985.

    Google Scholar 

  5. [5]

    F. M. de Oliveira Filho and F. Vallentin: Fourier analysis, linear programming, and densities of distance avoiding sets in Rn, Journal of the European Mathematical Society, 12 (2010), 1417–1428.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    K. Falconer: The realization of distances in measurable subsets covering Rn, Journal of Combinatorial Theory, Series A, 31 (1981), 184–189.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    K. Fischer: Additive k-colorable extensions of the rational plane, Discrete Mathematics 82 (1990), 181–195.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    D. Fisher and D. Ullman: The fractional chromatic number of the plane, Geombinatorics 2 (1992), 8–12.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    M. Gardner: Mathematical games, Scientific American, 206:172–180, 10 1960.

    Google Scholar 

  10. [10]

    J. Grytczuk, K. Junosza-Szaniawski, J. Sokół and K. Wesek: Fractional and j-fold colouring of the plane, Arxiv preprint:

  11. [11]

    H. Hadwiger: Ueberdeckung des Euklidischen Raumes durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.

    MathSciNet  MATH  Google Scholar 

  12. [12]

    R. Hochberg and P. O’Donnell: A large independent set in the unit distance graph, Geombinatorics 2 (1993), 83–84.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    G. M. Levin: Selected Topics in Fractional Graph Theory, 1997, Thesis (Ph.D.)–The Johns Hopkins University.

    Google Scholar 

  14. [14]

    S. L. Mahan: The fractional chromatic number of the plane, Master’s thesis, University of Colorado Denver, 1995.

    Google Scholar 

  15. [15]

    B. Mohar:

  16. [16]

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

    MATH  Google Scholar 

  17. [17]

    P. M. O’Donnell: High-girth unit-distance graphs, ProQuest LLC, Ann Arbor, MI, 1999, Thesis (Ph.D.)–Rutgers The State University of New Jersey-New Brunswick.

    Google Scholar 

  18. [18]

    D. Pritikin: All unit-distance graphs of order 6197 are 6-colorable, Journal of Combinatorial Theory, Series B 73 (1998), 159–163.

    MathSciNet  MATH  Google Scholar 

  19. [19]

    E. R. Scheinerman and D. H. Ullman: Fractional Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1997, A rational approach to the theory of graphs, with a foreword by Claude Berge, A Wiley-Interscience Publication.

    Google Scholar 

  20. [20]

    A. Soifer: The mathematical coloring book, Springer, New York, 2009, Mathematics of coloring and the colorful life of its creators, With forewords by Branko Grünbaum, Peter D. Johnson, Jr. and Cecil Rousseau.

    Google Scholar 

  21. [21]

    R. Solovay: A model of set-theory in which every set of reals is Lebesgue measurable, Annals of Mathematics (1970), 1–56.

    Google Scholar 

  22. [22]

    L. Székely: Measurable chromatic number of geometric graphs and sets without some distances in euclidean space, Combinatorica 4 (1984), 213–218.

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    D. Woodall: Distances realized by sets covering the plane, Journal of Combinatorial Theory, Series A 14 (1973), 187–200.

    MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Daniel W. Cranston.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cranston, D.W., Rabern, L. The fractional chromatic number of the plane. Combinatorica 37, 837–861 (2017).

Download citation

Mathematics Subject Classification (2000)

  • 05C15