Tur\'an number of complete bipartite graphs with bounded matching number

Let F be a family of graphs. A graph G is F-free if G does not contain any F∈ F as a subgraph. The Tur\'an number ex(n, F) is the maximum number of edges in an n-vertex F-free graph. Let Mₛ be the matching consisting of s independent edges. Recently, Alon and Frank determined the exact value of ex(n,ₘ,Mₛ₊₁\). Gerbner obtained several results about ex(n,,Mₛ₊₁\) when F satisfies certain proportions. In this paper, we determine the exact value of ex(n,,t,Mₛ₊₁\) when $s, n$ are large enough for every 3≤ l≤ t. When n is large enough, we also show that ex(n,2,2, Mₛ₊₁\)=n+s 2-/2 for s≥ 12 and ex(n,2,t,Mₛ₊₁\)=n+(t-1)s 2-/2 when t≥ 3 and s is large enough.