Let $f(n)$ be the maximum number of edges in a subgraph of $Q_n$ without a $C_4$, so that this conjecture is that $f(n)\leq (\frac{1}{2}+o(1))n2^{n-1}$.

Erdős [Er91] showed that\[f(n) \geq \left(\frac{1}{2}+\frac{c}{n}\right)n2^{n-1}\]for some constant $c>0$, and wrote it is 'perhaps not hopeless' to determine $f(n)$ exactly. Brass, Harborth, and Nienborg [BHN95] improved this to\[f(n) \geq \left(\frac{1}{2}+\frac{c}{\sqrt{n}}\right)n2^{n-1}\]for some constant $c>0$.

Balogh, Hu, Lidicky, and Liu [BHLL14] proved that $f(n)\leq 0.6068 n2^{n-1}$. This was improved to $\leq 0.60318 n2^{n-1}$ by Baber [Ba12b].

A similar question can be asked for other even cycles.

See also [666] and the entry in the graphs problem collection.

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This page was last edited 27 December 2025.