SOLVED

Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with
\[\geq \epsilon n2^{n-1}\]
many edges contains a $C_6$?

In [Er91] Erdős further suggests that perhaps, for every $k\geq 3$, there are constants $c$ and $a_k<1$ such that every subgraph with at least $cn^{a_k}2^n$ many edges contains a $C_{2k}$, where $a_k\to 0$ as $k\to \infty$.

The answer to this problem is no: Chung [Ch92] and Brouwer, Dejter, and Thomassen [BDT93] constructed an edge-partition of $Q_n$ into four subgraphs, each containing no $C_6$.

See also [86].