The generalizations of Erdős matching conjecture for $t$-matching number

Define a $t\mathit{\text{-matching}}$ of size $m$ in a $k$-uniform family as a collection $\{A_1,A_2,\dots ,A_m\}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ such that $\mathrm{\mid }{A}_i\cap A_j\mathrm{\mid }<t$ for all $1\le i<j\le m$. Let $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$. The $t\mathit{\text{-matching}}\mathit{\text{number}}$ 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 $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ with given $t$-matching number, which is a generalization of Erdős 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}.