Define a \textit{ $t$-matching} of size $m$ in a $k$-uniform family as a collection $\{ A_1, A_2, \ldots, A_m\} \subseteq \binom{ [n]}{ k}$ such that $|A_i \cap A_j| < t$ for all $1 \leq i < j \leq m$. Let $\mathcal{ F}\subseteq \binom{ [n]}{ k}$. The \textit{ $t$-matching number} of $\mathcal{ F}$, denoted by $\nu_t(\mathcal{ F})$, is the maximum size of a $t$-matching contained in $\mathcal{ F}$. We study the maximum cardinality of a family $\mathcal{ F}\subseteq\binom{ [n]}{ k}$ with given $t$-matching number, which is a generalization of Erd\H{ o}s matching conjecture, and we additionally prove a stability result. We also determine the second largest maximal structure with $\nu_t(\mathcal{ F})=s$, extending work of Frankl and Kupavskii \cite{ frankl2016two}. Finally, we obtain the extremal $G$-free induced subgraphs of generalized Kneser graph, generalizing Alishahi's results in \cite{ alishahi2018extremal}.
Categories: math.CO