Turán’s Theorem in the Hypercube

We are motivated by the analogue of Tura´n’s theorem in the hypercube $Q_n$: How many edges can a $Q_d$-free subgraph of $Q_n$ have? We study this question through its Ramsey-type variant and obtain asymptotic results. We show that for every odd $d$ it is possible to color the edges of $Q_n$ with $\frac{(d+1)^2}{4}$ colors such that each subcube $Q_d$ is polychromatic, that is, contains an edge of each color. The number of colors is tight up to a constant factor, as it turns out that a similar coloring with ${d+1\choose 2} +1$ colors is not possible. The corresponding question for vertices is also considered. It is not possible to color the vertices of $Q_n$ with $d+2$ colors such that any $Q_d$ is polychromatic, but there is a simple $d+1$ coloring with this property. A relationship to anti-Ramsey colorings is also discussed. We discover much less about the Tura´n-type question which motivated our investigations. Numerous problems and conjectures are raised.