Matchings in hypergraphs via Ore-degree conditions
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arXiv
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| Natura: | Preprint |
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2026
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| author | Balogh, József Palmer, Cory Raeisi, Ghaffar |
| author_facet | Balogh, József Palmer, Cory Raeisi, Ghaffar |
| contents | 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 ${σ_r}({\cal H})$. 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 $σ_r({\cal H})\leq r{n-2 \choose r-2}$, 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 $σ_r({\cal H})\leq r\left({n-2 \choose r-2}-{n-r-2 \choose r-2}\right)$.
(3) For $s\geq 2$ and $n\geq 3r^2(s-1)$, if ${\cal H}$ is an $r$-uniform hypergraph on $n$ vertices and $σ_r({\cal H})>r\left({n-1 \choose r-1}-{n-s \choose r-1}\right)$, then ${\cal H}$ contains $s$ pairwise disjoint edges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06415 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Matchings in hypergraphs via Ore-degree conditions Balogh, József Palmer, Cory Raeisi, Ghaffar Combinatorics 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 ${σ_r}({\cal H})$. 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 $σ_r({\cal H})\leq r{n-2 \choose r-2}$, 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 $σ_r({\cal H})\leq r\left({n-2 \choose r-2}-{n-r-2 \choose r-2}\right)$. (3) For $s\geq 2$ and $n\geq 3r^2(s-1)$, if ${\cal H}$ is an $r$-uniform hypergraph on $n$ vertices and $σ_r({\cal H})>r\left({n-1 \choose r-1}-{n-s \choose r-1}\right)$, then ${\cal H}$ contains $s$ pairwise disjoint edges. |
| title | Matchings in hypergraphs via Ore-degree conditions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.06415 |