Generalizations of the Erdős Matching Conjecture for the $t$-Matching Number

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 Erdős 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}.