Let $\mathcal{ H} \subseteq \binom{ [n]}{ r}$ be an $r$-uniform hypergraph on vertex set $[n] = \{ 1,2,\dots, n\}$. For an $r$-set of vertices $S \subseteq [n]$, the \emph{ degree} of $S$ is defined as $\textrm{ deg}(S)=\sum_{ v \in S}\textrm{ deg}(v)$ and the minimum of $\textrm{ deg}(S)$ over all non-edge $r$-subsets $S \not \in E(\mathcal{ H})$ of $V({ \cal H})$ is the { \it Ore-degree} of ${ \cal H}$, denoted by ${ \sigma_r}({ \cal H})$. We prove several Ore-degree results about existence of matchings in hypergraphs: (1) For $n\geq 2r+2$, if ${ \cal H}$ is an intersecting $r$-uniform hypergraph on $n$ vertices, then $\sigma_r({ \cal H})\leq r{ n-2 \choose r-2}$, and there is equality only when ${ \cal H}$ is a $1$-star. (2) For $r\geq 3$ and $n\geq 4r^2$, if is a non-trivial intersecting $r$-uniform hypergraph on $n$ vertices, then $\sigma_r({ \cal H})\leq r\left({ n-2 \choose r-2}-{ n-r-2 \choose r-2}\right)$. (3) For $s\geq 2$ and $n\geq 3r^2(s-1)$, if ${ \cal H}$ is an $r$-uniform hypergraph on $n$ vertices and $\sigma_r({ \cal H})>r\left({ n-1 \choose r-1}-{ n-s \choose r-1}\right)$, then ${ \cal H}$ contains $s$ pairwise disjoint edges.
Categories: math.CO