  A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of
vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemer\'edi
states that if $G$ is a graph of order $n$ with minimum degree $\delta(G) \ge
(t-1)n/t$ and $t| n$, then $G$ contains a perfect $K_t$-matching. Let $G$ be a
$t$-partite graph with vertex classes $V_1$,..., $V_t$ each of size $n$. We
show that if every vertex $x \in V_i$ is joined to at least $((t-1)/t +
\gamma)n $ vertices of $V_j$ for $i \ne j$, then $G$ contains a perfect
$K_t$-matching, thus verifying a conjecture of Fisher asymptotically.
Furthermore, we consider a generalisation to hypergraphs in terms of the
codegree.
