Chromatic numbers of algebraic hypergraphs

Abstract

Given a polynomial p(x 0,x 1,...,x k−1) over the reals ℝ, where each x i is an n-tuple of variables, we form its zero k-hypergraph H=(ℝn, E), where the set E of edges consists of all k-element sets {a 0,a 1,...,a k−1}⊆ℝn such that p(a 0,a 1,...,a k−1)=0. Such hypergraphs are precisely the algebraic hypergraphs. We say (as in [13]) that p(x 0,x 1,...,x k−1) is avoidable if the chromatic number χ(H) of its zero hypergraph H is countable, and it is κ-avoidable if χ(Hκ. Avoidable polynomials were completely characterized in [13]. For any infinite κ, we characterize the κ-avoidable algebraic hypergraphs. Other results about algebraic hypergraphs and their chromatic numbers are also proved.

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

References

  1. [1]

    B. Bukh: Measurable sets with excluded distances, Geom. Funct. Anal. 18 (2008), 668–697.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    J. Ceder: Finite subsets and countable decompositions of Euclidean spaces, Rev. Roumaine Math. Pures Appl. 14 (1969), 1247–1251.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    J. Fox: An infinite color analogue of Rado’s theorem, J. Combin. Theory, Ser. A 114 (2007), 1456–1469.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    R. L. Graham, B. L. Rothschild and Joel H. Spencer: Ramsey theory, 2nd edition, John Wiley & Sons, Inc., New York, 1990.

    Google Scholar 

  5. [5]

    P. Komjáth: Tetrahedron free decomposition of R 3, Bull. London Math. Soc. 23 (1991), 116–120.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    P. Komjáth: The master coloring, C. R. Math. Rep. Acad. Sci. Canada 14 (1992), 181–182.

    MathSciNet  MATH  Google Scholar 

  7. [7]

    P. Komjáth: A decomposition theorem for R n, Proc. Amer. Math. Soc. 120 (1994), 921–927.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    Péter Komjáth and James Schmerl: Graphs on Euclidean spaces defined using transcendental distances, Mathematika 58 (2012), 1–9.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    K. Kunen: Partitioning Euclidean space, Math. Proc. Cambridge Philos. Soc. 102 (1987), 379–383.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    J. H. Schmerl: Partitioning Euclidean space, Discrete Comput. Geom. 10 (1993), 101–106.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    J. H. Schmerl: Triangle-free partitions of Euclidean space, Bull. London Math. Soc. 26 (1994), 483–486.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    J. H. Schmerl: Countable partitions of Euclidean space, Math. Proc. Cambridge Philos. Soc. 120 (1996), 7–12.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    J. H. Schmerl: Avoidable algebraic subsets of Euclidean space, Trans. Amer. Math. Soc. 352 (2000), 2479–2489.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    A. Soifer: The mathematical coloring book, Mathematics of coloring and the colorful life of its creators, Springer, New York, 2009.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to James H. Schmerl.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Schmerl, J.H. Chromatic numbers of algebraic hypergraphs. Combinatorica 37, 1011–1026 (2017). https://doi.org/10.1007/s00493-016-3393-y

Download citation

Mathematics Subject Classification (2000)

  • 05C15
  • 05C63