Problem #714: Is it true that\[\mathrm{ex}(n; K_{r,r}) \gg n^{2-1/r}?

---

Progress

Kövári, Sós, and Turán [KST54] proved\[\mathrm{ex}(n; K_{r,r}) \ll n^{2-1/r}\]for all $r\geq 2$. Brown [Br66] and, independently, Erdős, Rényi, and Sós [ERS66], proved the conjectured lower bound when $r=3$.

When $r=2$ it is known that\[\mathrm{ex}(n;K_{2,2})=\left(\frac{1}{2}+o(1)\right)n^{3/2}\](see [768], since $K_{2,2}=C_4$).

See also [147].