Fischer's multipartite Hajnal–Szemerédi conjecture

Let $t\geq 3$ be an integer, and let $G$ be a balanced $t$-partite graph with vertex classes $V_1,\dots,V_t$, each of size $n$. For $1\leq i<j\leq t$, write $G[V_i,V_j]$ for the induced bipartite subgraph, and define

$$\widetilde{\delta}(G)=\min_{1\leq i<j\leq t}\delta(G[V_i,V_j]).$$

A perfect $K_t$-matching is a spanning set of vertex-disjoint copies of $K_t$ in $G$.

Fischer's conjecture. There exists an integer $a_{n,t}$ such that if

$$\widetilde{\delta}(G)\geq (t-1)n/t+a_{n,t},$$

then $G$ contains a perfect $K_t$-matching.

This conjecture is the multipartite analogue of the Hajnal–Szemerédi theorem, replacing a global minimum-degree condition by minimum-degree conditions between every pair of vertex classes. The paper verifies the conjecture asymptotically, showing that for every $\gamma>0$ the conclusion holds when the relevant pairwise minimum degrees are at least $((t-1)/t+\gamma)n$ and $n$ is sufficiently large.