A \(k\)-uniform hypergraph \(H\) is a pair \(((V(H), E(H))\) where \(V(H)\) is a finite set of vertices and \(E(H)\) is a family of \(k\)-element subsets of \(V(H)\). A matching of size \(s\) in \(H\) is a family of \(s\) pairwise disjoint edges of \(H\). Given a set \(S\subseteq V(H)\) the degree \(\deg(S)=\deg_H(S)\) is the number of edges of \(H\) containing \(S\).
The main result of this paper is this Ore-type theorem: Let \(n\), \(s\) be positive integers such that \(n\) is sufficiently large and \(n\geq 4s+7\). Suppose \(H\) is a 3-uniform hypergraph of order \(n\). If \(H\) contains no isolated vertices and \(\deg(u)+\deg(v)> 2\left(\binom{n-1}{2} -\binom{n-s}{2}\right)+1\) for any two vertices \(u\) and \(v\) contained in some edge of \(H\), then \(H\) either contains a matching of size \(s\) or is a subgraph of \(H_{n,s}\), where \(H_{n,s}\) is the \(3\)-uniform hypergraph of order \(n\) whose vertex-set is partitioned into two sets \(S\) and \(T\) of sizes \(n-2s+2\) and \(2s-1\), respectively, and whose edge-set consists of all triples with at least two vertices in \(T\). This result improves an earlier one by the same authors [ibid. 342, No. 6, 1731–1737 (2019; Zbl 1414.05244)].