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arXiv · 2508.12679

Generalizations of the Erd\H{o}s Matching Conjecture for the $t$-Matching Number

Mengyu Cao·Mei Lu·Haixiang Zhang
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Abstract

We write finite set systems as uniform hypergraphs. A \emph{$t$-matching} in a $k$-uniform hypergraph is a set of hyperedges any two of which intersect in fewer than $t$ vertices. The maximum size of such a set is the \emph{$t$-matching number} and is denoted by $\nu_t$. We study the maximum number of hyperedges in a $k$-uniform hypergraph on $[n]$ with prescribed $t$-matching number. This gives a hypergraph analogue of the Erd\H{o}s Matching Conjecture. We also determine the second largest maximal structure with $\nu_t(\mathcal{F})=s$, extending work of Frankl and Kupavskii \cite{frankl2016two}. And, we obtain the extremal $G$-free induced subgraphs of generalized Kneser graph, generalizing Alishahi's results in \cite{alishahi2018extremal}.

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Mengyu Cao, Mei Lu, Haixiang Zhang. 2025-08-18. Generalizations of the Erd\H{o}s Matching Conjecture for the $t$-Matching Number. https://arxiv.org/abs/2508.12679

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