Equitable colorings of non-uniform simple hypergraphs

The paper is devoted to the combinatorial problem concerning equitable colorings of non-uniform simple hypergraphs. Let H=(V,E)H=(V,E) be a hypergraph, a coloring with r colors of its vertex set V is called equitable if it is proper (i.e. none of the edges is monochromatic) and the cardinalities of the color classes differ by at most one. We show that if H is a simple hypergraph with minimum edge-cardinality n and∑e∈Er1−|e|≥cn, for some absolute constant c>0c>0, then H has an equitable r-coloring.