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 the property B problem Radhakrishnan and Srinivasan proved that if $H$ is a $k$-uniform hypergraph with maximum vertex degree $\Delta \left(H\right)$ satisfying $\Delta \left(H\right)⩽c\frac{2^{k-1}}{\sqrt{klnk}}$ 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 or has a lot of vertices, then under the same condition on the maximum vertex degree it has an equitable coloring with two colors. We also obtain a general result for equitable colorings of partial Steiner systems.