Color-detectors of hypergraphs

Abstract

Let X be a set of cardinality k, F be a family of subsets of X. We say that a cardinal λ,λ<k, is a color-detector of the hypergraph H=(X,F) if card χ(X)≤λ for every coloring χ:X→k such that card χ(F)≤λ for every F∈F. We show that the color-detectors of H are tightly connected with the covering number cov(H)=sup{α: any α points of X are contained in some F∈F}. In some cases we determine all of the color-detectors of H and their asymptotic counterparts. We put also some open questions.