Chromatic numbers of algebraic hypergraphs

Abstract

Given a polynomial p(x0,x1,...,xk−1) over the reals R, where each xi 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 {a0,a1,...,ak−1}⊆Rn such that p(a0,a1,...,ak−1)=0. Such hypergraphs are precisely the algebraic hypergraphs. We say (as in [13]) that p(x0,x1,...,xk−1) is avoidable if the chromatic number χ(H) of its zero hypergraph H is countable, and it is κ-avoidable if χ(H≤κ. 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.