Equitable two-colorings of uniform hypergraphs

An equitable two-coloring of a hypergraph H = (V, E) is a proper vertex two-coloring such that the cardinalities of color classes differ by at most one. In connection with property B problem Radhakrishnan and Srinivasan proved that if H is a k-uniform hypergraph with maximum vertex degree ∆(H) satisfying ∆(H) c 2 k−1 √ k ln k (1) for some absolute constant c > 0, then H is 2-colorable. By using the Lovász Local Lemma for negatively correlated events and the random recoloring method we prove that if H either is a simple hypergraph (every two distinct edges have at most one common vertex) or has a lot of vertices, then under the same condition (1) on the maximum vertex degree it has an equitable coloring with two colors.