Matchings in hypergraphs via Ore-degree conditions

JÃ³zsef Balogh; Cory Palmer; Ghaffar Raeisi

paper

Matchings in hypergraphs via Ore-degree conditions

arXiv:2603.06415

Abstract

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

Update to the statement of Conjecture 2