Turan number of complete bipartite graphs with bounded matching number

Let F be a family of graphs. A graph Gis F-free if Gdoes not contain any F is an element of F as a subgraph. The Turan number ex(n, F) is the maximum number of edges in an n-vertex F-free graph. Let Msbe the matching consisting of sindependent edges. Recently, Alon and Frankl determined the exact value of ex(n,{K-m, Ms+1}). Gerbner obtained several results about ex(n,{F, Ms+1}) when Fsatisfies certain properties. In this paper, we determine the exact value of ex(n,{K-l,K-t, Ms+1}) when s, nare large enough for every 3 <= l <= t. When nis large enough, we also show that ex(n,{K-2,K-2, Ms+1})= n+ ((s)(2)) - inverted right perpendicular s/2 inverted left perpendicular for s= 12and ex(n,{K-2,K- t, Ms+1})= n+(t-1)((s)(2)) - inverted right perpendicular s/2 inverted left perpendicular when t >= 3 and sis large enough. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.