Equitable colorings of hypergraphs with few edges

The paper deals with an extremal problem concerning equitable colorings of uniform hypergraph. Recall that a vertex coloring of a hypergraph H is called proper if there are no monochromatic edges under this coloring. A hypergraph is said to be equitably r-colorable if there is a proper coloring with r colors such that the sizes of any two color classes differ by at most one. In the present paper we prove that if the number of edges vertical bar E(H)vertical bar <= 0.01 (n/ln n)(r-)(1/r) rn-1 then the hypergraph H is equitably r-colorable provided r < (5)root ln n. (C) 2019 Elsevier B.V. All rights reserved.