Perfect Matchings in 4-uniform hypergraphs

A perfect matching in a 4-uniform hypergraph is a subset of $\lfloor\frac{n}{4}\rfloor$ disjoint edges. We prove that if $H$ is a sufficiently large 4-uniform hypergraph on $n=4k$ vertices such that every vertex belongs to more than ${n-1\choose 3} - {3n/4 \choose 3}$ edges then $H$ contains a perfect matching. This bound is tight and settles a conjecture of H{á}n, Person and Schacht.

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