Color-detectors of hypergraphs

Abstract

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