A multipartite version of the Hajnal-Szemer\'edi theorem for graphs and hypergraphs

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$G$containsaperfect$K_t$-matching\mathrm{.}Let$G$bea$t$-partitegraphwithvertexclasses$V_1$,\mathrm{.}\mathrm{.}\mathrm{.},$V_t$eachofsize$n$. We show that if every vertex $x\in V_i$ is joined to at least $((t-1)/t + \gamma)n $verticesof$V_j$for$i \ne j$,then$G$containsaperfect$K_t$-matching, thus verifying a conjecture of Fisher asymptotically. Furthermore, we consider a generalisation to hypergraphs in terms of the codegree.