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


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

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AMS subject classification (1980)

  • 05 C 15
  • 52 A 45