On degree bounds of $k$-uniform hypergraphs with bounded matching number

TL;DR

This paper establishes degree bounds in $k$-uniform hypergraphs that guarantee large matchings, generalizing previous theorems and optimizing conditions on the number of vertices.

Contribution

It extends existing results by providing new degree bounds that ensure large matchings in hypergraphs, including optimal bounds for the number of vertices.

Findings

Proves that a degree condition on the $(2sk+1)$-th largest degree guarantees a matching of size at least $s$.

Generalizes and improves bounds related to Ore-degree conditions for matchings.

Shows the bounds on the number of vertices can be linear in $sk$, confirming a conjecture.

Abstract

We study the connection between the degree sequence of a $k$-uniform hypergraph and the size of its largest matching. Let $\mathcal{F}$ be a $k$-uniform hypergraph on $n$ vertices and let $d_1 \ge d_2 \ge \dots \ge d_n$ be the vertex degrees arranged in non-increasing order. For integers $k\ge 2$, $s\ge 2$ and $n > 2sk$, we prove that if the $(2sk+1)$-th largest degree satisfies $d_{2sk+1} > \binom{n-1}{k-1} - \binom{n-s}{k-1},$ then $\mathcal{F}$ contains a matching of size at least $s$. This can be viewed as a generalization of theorems by Lu, Guo, and Jiang (2023) and Huang and Rao (2026). Moreover, by relaxing the range of $n$, we obtain the same bound for the $(k+2s-2)$-th largest degree vertex. Note that the number $k+2s-2$ is optimal. For a $k$-set of vertices $S \subseteq [n]$, the degree of $S$ is defined as $\mathrm{deg}(S) = \sum_{v \in S} \mathrm{deg}(v)$, and the minimum of $\mathrm{deg}(S)$ over all non-edge $k$-subsets $S \notin E(\mathcal{F})$ of $V(\mathcal{F})$ is the Ore-degree of $\mathcal{F}$, denoted by $\sigma_k(\mathcal{F})$. Balogh, Palmer and Raeisi proved: for $s \ge 2$ and $n \ge 3k^2(s-1)$, if $\sigma_k(\mathcal{F}) > k\left(\binom{n-1}{k-1} - \binom{n-s}{k-1}\right),$ then $\mathcal{F}$ contains a matching of size $s$. They also conjectured that the result holds when $n > sk$. As a corollary, we prove that the bound on $n$ can be taken to be linear in $sk$ ($ n \geq 3sk $).