Color-detectors of hypergraphs

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 $cov(H)=\mathrm{cup}\{\alpha:\text{any }\alpha\text{points of }X\text{ are contained in some }F\in\mathcal F\}$. In some cases we determine all of the color-detectors of $H$ and their asymptotic counterparts. We put also some open questions.