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 Mₒ be the matching consisting of s independent edges. Recently, Alon and Frank determined the exact value of ex (n, \K₌, Mₒ+₁\). Gerbner obtained several results about ex (n, \F, Mₒ+₁\) when F satisfies certain proportions. In this paper, we determine the exact value of ex (n, \K₋, ₓ, Mₒ+₁\) when s, n are large enough for every 3 l t. When n is large enough, we also show that ex (n, \K₂, ₂, Mₒ+₁\) =n+s 2-2 for s 12 and ex (n, \K₂, ₓ, Mₒ+₁\) =n+ (t-1) s 2-2 when t 3 and s is large enough.