OPEN
This is open, and cannot be resolved with a finite computation.
Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?
Conjectured by Erdős and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.
The behaviour of $\mathrm{ex}(n;C_4)$ is the subject of
[765].
He, Ma, and Yang
[HeMaYa21] have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.
View the LaTeX source
This page was last edited 18 November 2025.
Additional thanks to: Desmond Weisenberg
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #60, https://www.erdosproblems.com/60, accessed 2026-01-16
