Geodesic
Color-detectors of hypergraphs
Algebra and discrete mathematics, no. 1 (2005), pp. 84-91

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Let $X$ be a set of cardinality $k$, $\mathcal{F}$ be a family of subsets of $X$. We say that a cardinal $\lambda,\lambda$, 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.
Keywords: hypergraph, color-detector, covering number. 
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     author = {I. V. Protasov and O. I. Protasova},
     title = {Color-detectors of hypergraphs},
     journal = {Algebra and discrete mathematics},
     pages = {84--91},
     publisher = {mathdoc},
     number = {1},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2005_1_a7/}
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I. V. Protasov; O. I. Protasova. Color-detectors of hypergraphs. Algebra and discrete mathematics, no. 1 (2005), pp. 84-91. http://geodesic.mathdoc.fr/item/ADM_2005_1_a7/