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arXiv · 2601.04929

Stability results for Berge-matching in hypergraphs

Jia-Bao Yang·Leilei Zhang
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Abstract

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 Tur\'an 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 $M_k$, 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 Erd\H{o}s-Gallai theorem (Erd\H{o}s and Gallai, 1959 [5]).

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Jia-Bao Yang, Leilei Zhang. 2026-01-08. Stability results for Berge-matching in hypergraphs. https://arxiv.org/abs/2601.04929

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