On colorings of non-uniform hypergraphs without short cycles

The work deals with a generalization of Erdős – Lovász problem concerning colorings of non-uniform hypergraphs. Let H = ( V , E ) be a hypergraph and let f r ( H ) = ∑ e ∈ E r 1 − | e | for some r ⩾ 2 . Erdős and Lovász asked about 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 our paper we study similar problem in the class of hypergraphs with large girth.