The generalizations of Erdős matching conjecture for t-matching number

Define a t-matching of size m in a k-uniform family as a collection \A₁, A₂, , Aₘ\ nk such that |Aᵢ Aⱼ| < t for all 1 i < j m. Let F nk. The t-matching number of F, denoted by νₜ (F), is the maximum size of a t-matching contained in F. We study the maximum cardinality of a family Fk 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 νₜ (F) =s, extending work of Frankl and Kupavskii frankl2016two. Finally, we obtain the extremal G-free induced subgraphs of generalized Kneser graph, generalizing Alishahi's results in alishahi2018extremal.