Erd s–Simonovits even-cycle extremal-number conjecture

Let $\ell\geqslant2$, and write

$$\operatorname{ex}\bigl(n,\{C_3,C_4,\ldots,C_{2\ell}\}\bigr)$$

for the maximum number of edges in an $n$-vertex graph containing none of the cycles $C_3,C_4,\ldots,C_{2\ell}$. Erd s–Simonovits conjecture.

$$\operatorname{ex}\bigl(n,\{C_3,C_4,\ldots,C_{2\ell}\}\bigr)=\Theta\bigl(n^{1+1/\ell}\bigr).$$

The source invokes this conjecture conditionally to show that the bounds of its random Tur n theorem are essentially best possible.