Matchings in hypergraphs via Ore-degree conditions

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.