The generalizations of Erd\H{o}s matching conjecture for $t$-matching number

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}.