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 $\left(\genfrac{}{}{0px}{}{n-1}{3}\right)-\left(\genfrac{}{}{0px}{}{3n\mathrm{/}4}{3}\right)$ edges then $H$ contains a perfect matching. This bound is tight and settles a conjecture of H{'a}n, Person and Schacht.