$K_{2,t+1}$-free graphs with many copies of $K_{t,t}$

Abstract

For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $Ω_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[ \mathrm{ex}(n,K_{t,t},K_{2,t+1})=Θ_t(n^2). \] This answers a question of Spiro.