$K_{2,t+1}$-free graphs containing an optimal number of $K_{t,t}$'s

Abstract

The generalized Turán number $ex(n,K_{t,t},K_{2,t+1})$ is the maximum number of copies of $K_{t,t}$ that a $K_{2,t+1}$-free graph on $n$ vertices can contain. Recently, Pohoata, Tidor, and Yu established that $ex(n,K_{t,t},K_{2,t+1})={\Theta }_t(n^2)$ for all integers $t\ge 3$. In this short note, we use an explicit construction to establish that when $t$ is a prime power and $n=t^{2e-1}$, then $$ex(n,K_{t,t},K_{2,t+1})=(1+o(1))\frac{n^2}{2t(t-1)}.$$

6 pages