Equitable colorings of hypergraphs with few edges

Abstract 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 | E ( H ) | ≤ 0 . 01 n ln n r − 1 r r n − 1 then the hypergraph H is equitably r -colorable provided r ln n 5 .

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