Coloring Non-uniform Hypergraphs Without Short Cycles

The work deals with a generalization of Erdos-Lovasz problem concerning colorings of non-uniform hypergraphs. Let H = (V, E) be a hypergraph and let f(r)(H) = Sigma(e is an element of E)r(1-vertical bar e vertical bar) for some r >= 2. Erdos and Lovasz proposed to find the value f(n) equal to the minimum possible value of f(2)(H) where H is 3-chromatic hypergraph with minimum edge-cardinality n. In the paper we study similar problem for the class of hypergraphs with large girth. We prove that if H is a hypergraph with minimum edge-cardinality n >= 3 and girth at least 4, satisfying the inequality f(r)(H) <= 1/2 (n/ln n)(2/3), then H is r-colorable. Our result improves previous lower bounds for f (n) in the class of hypergraphs without 2- and 3-cycles.