On harmonious coloring of hypergraphs

A harmonious coloring of a k-uniform hypergraph H is a vertex coloring such that no two vertices in the same edge have the same color, and each k-element subset of colors appears on at most one edge. The harmonious number h (H) is the least number of colors needed for such a coloring. The paper contains a new proof of the upper bound h (H) =O (k!m) on the harmonious number of hypergraphs of maximum degree with m edges. We use the local cut lemma of A. Bernshteyn.