Equitable colorings of hypergraphs with $r$ colors

The paper deals with a problem concerning equitable vertex colorings of uniform hypergraphs, i.e., colorings under which there are no monochromatic edges and simultaneously all the color classes have almost the same cardinalities. We obtain a new bound on the edge number of an $n$-uniform hypergraph that guarantees the existence of an equitable vertex coloring with $r$ colors for this hypergraph.