Coloring 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 fr(H)=∑e∈Er1-|e|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{f_r(H)=\sum\limits_{e \in E}r^{1-|e|}}}$$\end{document} for some r ≥ 2. Erdős and Lovász proposed to find the value f (n) equal to the minimum possible value of f2(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 fr(H)≤12nlnn2/3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_r(H) \leq \frac{1}{2}\, \left(\frac{n}{{\rm ln}\, n}\right)^{2/3},$$\end{document}then H is r -colorable. Our result improves previous lower bounds for f (n) in the class of hypergraphs without 2- and 3-cycles.