Turán's theorem in the hypercube

We are motivated by the analogue of Turán'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 (d=1)²/4 colors such that each subcube Qd 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 (₂^d+1) + 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 Turán-type question which motivated our investigations. Numerous problems and conjectures are raised.