Turán number of an induced complete bipartite graph plus an odd cycle

Abstract

Let $k\ge 2$ be an integer. We show that if $s=2$ and $t\ge 2$, or $s=t=3$, then the maximum possible number of edges in a $C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$ is asymptotically equal to $(t-s+1{)}^{1/s}{\left(\frac{n}{2}\right)}^{2-1/s}$ except when $k=s=t=2$. This strengthens a result of Allen, Keevash, Sudakov and Verstraëte and answers a question of Loh, Tait and Timmons.