A note on two-colorability of nonuniform hypergraphs

For a hypergraph $H$, let $q(H)$ denote the expected number of monochromatic edges when the color of each vertex in $H$ is sampled uniformly at random from the set of size 2. Let $s_{\min }(H)$ denote the minimum size of an edge in $H$. Erd\H{o}s asked in 1963 whether there exists an unbounded function $g(k)$ such that any hypergraph $H$ with $s_{\min }(H)\ge k$ and $q(H)\le g(k)$ is two colorable. Beck in 1978 answered this question in the affirmative for a function $g(k)=\mathrm{\Theta }({\log }^{\ast }k)$. We improve this result by showing that, for an absolute constant $\delta >0$, a version of random greedy coloring procedure is likely to find a proper two coloring for any hypergraph $H$ with $s_{\min }(H)\ge k$ and $q(H)\le \delta \cdot \log k$.