A stability result for almost perfect matchings

Let n,k,s be three integers and β be a sufficiently small positive number such that k≥ 3, 0<1/n≪β≪ 1/k and ks+k≤ n≤ (1+β)ks+k-2. A k-graph is called non-trivial if it has no isolated vertex. In this paper, we determine the maximum number of edges in a non-trivial k-graph with n vertices and matching number at most s. This result confirms a conjecture proposed by Frankl (On non-trivial families without a perfect matching, European J. Combin., 84 (2020), 103044) for the case when s is sufficiently large.