Matchings in hypergraphs via Ore-degree conditions

Let $\mathcal{H}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{r}\right)$ 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 $\text{deg}(S)={\sum }_{v\in S}\text{deg}(v)$ and the minimum of $\text{deg}(S)$ over all non-edge $r$-subsets $S\in ̸E(\mathcal{H})$ of $V(\mathcal{H})$ is the {\it Ore-degree} of $\mathcal{H}$, denoted by ${\sigma }_r(\mathcal{H})$. We prove several Ore-degree results about existence of matchings in hypergraphs: (1) For $n\ge 2r+2$, if $\mathcal{H}$ is an intersecting $r$-uniform hypergraph on $n$ vertices, then ${\sigma }_r(\mathcal{H})\le r\left(\genfrac{}{}{0px}{}{n-2}{r-2}\right)$, and there is equality only when $\mathcal{H}$ is a $1$-star. (2) For $r\ge 3$ and $n\ge 4r^2$, if is a non-trivial intersecting $r$-uniform hypergraph on $n$ vertices, then ${\sigma }_r(\mathcal{H})\le r(\left(\genfrac{}{}{0px}{}{n-2}{r-2}\right)-\left(\genfrac{}{}{0px}{}{n-r-2}{r-2}\right))$. (3) For $s\ge 2$ and $n\ge 3r^2(s-1)$, if $\mathcal{H}$ is an $r$-uniform hypergraph on $n$ vertices and ${\sigma }_r(\mathcal{H})>r(\left(\genfrac{}{}{0px}{}{n-1}{r-1}\right)-\left(\genfrac{}{}{0px}{}{n-s}{r-1}\right))$, then $\mathcal{H}$ contains $s$ pairwise disjoint edges.