Turán number for cubes

For a finite family \( {\mathcal F}\) of graphs, let \( t(n, {\mathcal{F}})\) denote the smallest integer \( m\) that every graph on \( n\) vertices and \( m\) edges must contain a member of \( \mathcal{F}\) as a subgraph.

A problem on Turán numbers for an \(n\)-cube} (proposed by Erdös and Simonovits[1], 1970)

Let \(Q_k\) denote the \(k\)-cube on \(2^k\) vertices. Determine \(t(n,Q_k)\).
In particular, determine \(t(n,Q_3)\).

Erdös and Simonovits[1] proved that

\(\displaystyle t(n,Q_3) \leq c n^{8/5}. \)

An obvious lower bound for \( t(n,Q_3)\) is \( t(n,C_4) = (\frac 1 2+o(1)) n^{3/2}\). However, no better lower bound than this is known.

1 P. Erdös and M. Simonovits, Some extremal problems in graph theory, Combinatorial theory and its applications, I (Proc. Colloq., Balatonfüred, 1969), 377-390, North-Holland, Amsterdam, 1970.