Ore-degree conditions for hypergraph matchings

Abstract

Let ${\sigma }_k(H)$ be the minimum sum of the vertex degrees over all non-edges of an $n$-vertex $k$-uniform hypergraph $H$. For every $k\ge 3$, $n\ge ks$ and sufficiently large $n$, we establish a sharp Ore-type degree condition that forces a matching of size $s$ in $H$. Specifically, we prove that if \[ σ_k(H)>\max\left\{ k\left(\binom{n-1}{k-1}-\binom{n-s}{k-1}\right), (k-c)\binom{ks-2}{k-1}\right\}, \] then $H$ has a matching of size $s$, where $c=n-ks+1$. In particular, this proves a conjecture of Balogh, Palmer and Raeisi for $k\ge 3$ and sufficiently large $n$.

arXiv comment19 pages