Measurable chromatic number of geometric graphs and sets without some distances in euclidean space

Abstract

LetH be a set of positive real numbers. We define the geometric graphG H as follows: the vertex set isR n (or the unit circleS 1) andx, y are joined if their distance belongs toH. We define the measurable chromatic number of geometric graphs as the minimum number of classes in a measurable partition into independent sets.

In this paper we investigate the difference between the notions of the ordinary and measurable chromatic numbers. We also prove upper and lower bounds on the Lebesgue upper density of independent sets.

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References

  1. [1]

    N. G. de Bruijn andP. Erdös. A colour problem for infinite graphs and a problem in theory of relations,Nederl. Akad. Wetensch. Proc. Ser. A.54 (1951), 371–363.

    MATH  Google Scholar 

  2. [2]

    H. T. Croft, Incidence incidents,Eureka (Cambridge)30 (1967), 22–26.

    Google Scholar 

  3. [3]

    P. Erdös andM. Simonovits, On the cromatic number of geometric graphs,Ars Combinatoria.9 (1980), 229–246.

    MATH  MathSciNet  Google Scholar 

  4. [4]

    K. F. Falconer, The realization of distances in measurable subsets coveringRn.J. Comb. Theory A31, 187–189.

  5. [5]

    P. Frankl andR. M. Wilson, Intersection theorems with geometric consequences,Combinatorica1 (4) (1981), 357–368.

    MATH  MathSciNet  Google Scholar 

  6. [6]

    H. Hadwiger, Ungelöste Probleme No. 40,Elemente der Math.16 (1961), 103–104.

    MathSciNet  Google Scholar 

  7. [7]

    D. G. Larman andC. A. Rogers, The realization of distances within sets in Euclidean space,Mathematika19 (1972), 1–24.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    L. Moser andW. Moser, Solution to Problem 10,Canad. Math. Bull.4 (1961), 187–189.

    Google Scholar 

  9. [9]

    W. Moser,Research problems in discrete geometry, (mimeographed), 1981.

  10. [10]

    D. R. Woodall, Distances realized by sets covering the plane,J. Combinatorial Theory A14 (1973), 187–200.

    MATH  Article  MathSciNet  Google Scholar 

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Székely, L.A. Measurable chromatic number of geometric graphs and sets without some distances in euclidean space. Combinatorica 4, 213–218 (1984). https://doi.org/10.1007/BF02579223

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AMS subject classification (1980)

  • 05 C 15
  • 52 A 45