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Semi-Strong Coloring of Intersecting Hypergraphs

For any c >= 2, a c-strong coloring of the hypergraph G is an assignment of colors to the vertices of G such that for every edge e of G, the vertices of e are colored by at least min{c,|e|} distinct colors. The hypergraph G is t-intersecting if every two edges of G have at least t vertices in common. We ask: for fixed c >= 2 and t >= 1, what is the minimum number of colors that is sufficient to c-strong color any t-intersecting hypergraphs? The purpose of this note is to answer the question for some values of t and c and, more importantly, to describe the settings for which the question is still open. We show that when t <= c-2, no finite number of colors is sufficient to c-strong color all t-intersecting hypergraphs. It is still unknown whether a finite number of colors suffices for the same task when t = c-1 and c > 2. In the last case, when t >= c, we show with a probabilistic argument that a finite number of colors is sufficient to c-strong color all t-intersecting hypergraphs, but a large gap still remains between the best upper and lower bounds on this number.