## 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(*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.

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

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### Mathematics Subject Classification (2000)

- 05C15
- 05C63