  Let $q$ be a positve integer, and $G$ be a $q$-partite simple graph on $qn$
vertices, with $n$ vertices in each vertex class. Let $\delta={k_q \over
k_q+1}$, where $k_q=q+O(\log{q})$. If each vertex of $G$ is adjacent to at
least $\delta n$ vertices in each of the other vertex classes, $q$ is bounded
and $n$ is large enough, then $G$ has a $K_q$-factor.
