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 withcolors, 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 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 Turán-type question which motivated our investigations. Numerous problems and conjectures are raised.