  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 $s_{\min}(H) \geq k$ and $q(H) \leq g(k)$ is two
colorable. Beck in 1978 answered this question in the affirmative for a
function $g(k) = \Theta(\log^* k)$. 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
$s_{\min}(H) \geq k$ and $q(H) \leq \delta \cdot \log k$.
