Stability results for Berge-matching in hypergraphs

Given a graph F , a hypergraph is called a Berge- F if it can be obtained by expanding each edge of F into a hyperedge containing it. Let M k denote the matching of size k . Kang, Ni, and Shan [12] determined the Turán number of Berge- M k . Our main result shows that if an r -uniform hypergraph H on n vertices has nearly as many edges as the extremal in their theorem without containing Berge- M k , then H must be structurally close to certain well-specified graphs. Meanwhile, our result also implies several stability results, such as the stability version of the well-known Erdős-Gallai theorem (Erdős and Gallai, 1959 [5] ).