Problem Statement
Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?
Categories:
Graph Theory Cycles
Progress
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 $q$.
Source: erdosproblems.com/60 | Last verified: January 13, 2026