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 , < k, is a color-detector of the hypergraph H = (X,F) if card (X) for every coloring : X ! k such that card (F) for every F 2 F. We show that the color-detectors of H are tightly connected with the covering number cov (H) = sup{ : any points of X are contained in some F 2 F}. In some cases we determine all of the color-detectors of H and their asymptotic counterparts. We put also some open questions.