The asymptotic extremal-number conjecture for $K_{t,t}$ in $K_{2,t+1}$-free graphs

Let $t$ be a prime power, and let $ex(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex graph containing no copy of $F$. In particular, consider $ex(n,K_{t,t},K_{2,t+1})$.

Asymptotic extremal-number conjecture.

$$ex(n, K_{t, t}, K_{2, t+1}) = (1 + o(1))\frac{n^2}{2t(t-1)}.$$

The paper proves the matching lower bound up to a factor of $1+o(1)$ and obtains an upper bound from an earlier proposition; the conjecture asserts that this upper bound gives the true asymptotic growth.