Near perfect matchings in uniform hypergraphs

Abstract

In this paper, we study degree conditions for the existence of large matchings in uniform hypergraphs. We prove that for integers $k,l,n$ with $k\ge 3$, $k/2<l<k$, and $n$ large, if $H$ is a $k$-uniform hypergraph on $n$ vertices and ${\delta }_l(H)>\left(\genfrac{}{}{0ex}{}{n-l}{k-l}\right)-\left(\genfrac{}{}{0ex}{}{(n-l)-(\lceil n/k\rceil -2)}{2}\right)$, then $H$ has a matching covering all but a constant number of vertices. When $l=k-2$ and $k\ge 5$, such a matching is near perfect and our bound on ${\delta }_l(H)$ is best possible. When $k=3$, with the help of an absorbing lemma of Hán, Person, and Schacht, our proof also implies that $H$ has a perfect matching, a result proved by K\" uhn, Osthus, and Treglown and, independently, of Kahn.

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