The shifting method and generalized Turán number of matchings

Given two graphs T and F, the maximum number of copies of T in an F-free graph on n vertices is called the generalized Turán number, denoted by ex(n,T,F). When T=K2, it reduces to the classical Turán number ex(n,F). Let Mk be a matching with k edges and Ks,t∗ a graph obtained from Ks,t by replacing the part of size s by a clique of the same size. In this paper, we show that for any s≥2 and n≥2k+1, ex(n,Ks,Mk+1)=max2k+1s,ks+(n−k)ks−1. For any...