On the -strong chromatic number of -intersecting hypergraphs

For a fixed c ≥ 2 , a c -strong coloring of the hypergraph G is a vertex coloring such that each edge e of G covers vertices with at least min { c , | e | } distinct colors. A hypergraph is t -intersecting if the intersection of any two of its edges contains at least t vertices. This paper addresses the question: what is the minimum number of colors which suffices to c -strong color any t -intersecting hypergraph? We first show that the number of colors required to c -strong color a hypergraph of size n is O ( n ) . Then we prove that we can use finitely many colors to 3-strong color any 2-intersecting hypergraph. Finally, we show that 2 c − 1 colors are enough to c -strong color any shifted ( c − 1 ) -intersecting hypergraph, and 2 c − 2 colors are enough to c -strong color any shifted t -intersecting hypergraph for t ≥ c . Both chromatic numbers are optimal and match conjectured statements in which the shifted condition is removed.