OPEN

Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of
\[\mathrm{ex}(n;Q_k).\]

Erdős and Simonovits [ErSi70] proved that
\[(\tfrac{1}{2}+o(1))n^{3/2}\leq \mathrm{ex}(n;Q_3) \ll n^{8/5}.\]
A theorem of Sudakov and Tomon [SuTo22] implies
\[\mathrm{ex}(n;Q_k)=o(n^{2-\frac{1}{k}}).\]
Janzer and Sudakov [JaSu22b] have improved this to
\[\mathrm{ex}(n;Q_k)\ll_k n^{2-\frac{1}{k-1}+\frac{1}{(k-1)2^{k-1}}}.\]
See also the entry in the graphs problem collection.