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 ₁, A₂, …, Aₘ\ ⊆ [n]k such that |Aᵢ ∩ Aⱼ| < t for all 1 ≤ i < j ≤ m. Let F⊆ [n]k. 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 F⊆[n]k 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}.