  The smallest number of edges forming an n-uniform hypergraph which is not
r-colorable is denoted by m(n,r). Erd\H{o}s and Lov\'{a}sz conjectured that
m(n,2)=\theta(n 2^n)$. The best known lower bound m(n,2)=\Omega(sqrt(n/log(n))
2^n) was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple
proof of their result. The proof is based on analysis of random greedy coloring
algorithm investigated by Pluh\'ar in 2009. The proof method extends to the
case of r-coloring, and we show that for any fixed r we have
m(n,r)=\Omega((n/log(n))^(1-1/r) r^n) improving the bound of Kostochka from
2004. We also derive analogous bounds on minimum edge degree of an n-uniform
hypergraph that is not r-colorable.
