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Combinatorica

, Volume 37, Issue 5, pp 1011–1026 | Cite as

Chromatic numbers of algebraic hypergraphs

  • James H. SchmerlEmail author
Original Paper

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=(ℝ n , E), where the set E of edges consists of all k-element sets {a 0,a 1,...,a k−1}⊆ℝ n 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.

Mathematics Subject Classification (2000)

05C15 05C63 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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