Coloring Non-uniform Hypergraphs Without Short Cycles

The work deals with a generalization of Erd?s–Lovász problem concerning colorings of non-uniform hypergraphs. Let H  = (V, E) be a hypergraph and let ({{f_r(H)=sumlimits_{e in E}r^{1-|e|}}}) for some r ≥ 2. Erd?s and Lovász 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 $$f_r(H) leq frac{1}{2}, left(frac{n}{{rm ln}, n}right)^{2/3},$$ then H is r -colorable. Our result improves previous lower bounds for f (n) in the class of hypergraphs without 2- and 3-cycles.