Perfect matching in 4-partite 4-uniform hypergraphs

A balanced $k$-partite $k$-graph is a $k$-uniform hypergraph such that every edge intersects each partition class in exactly one vertex, where each partition class has size $n$. Lo and Markstr\"om (2014) determined the minimum vertex-degree threshold for perfect matchings in balanced $3$-partite $3$-graphs. In this paper, we determine the minimum vertex-degree threshold for balanced $4$-partite $4$-graphs. The proof relies on a reduction framework for $k$-partite $k$-graphs, through which the existence of a perfect fractional matching is converted into a finite-dimensional optimization problem.