Vertex Turán problems in the hypercube

Let Qn be the n-dimensional hypercube: the graph with vertex set n{0,1} and edges between vertices that differ in exactly one coordinate. For 1⩽d⩽n and F⊆d{0,1} we say that S⊆n{0,1} is F-free if every embedding satisfies i(F)⊈S. We consider the question of how large S⊆n{0,1} can be if it is F-free. In particular we generalise the main prior result in this area, for F=2{0,1}, due to E.A. Kostochka and prove a local stability result for the structure of near-extremal sets.We also show that the density required to guarantee an embedded copy of at least one of a family of forbidden configurations may be significantly lower than that required to ensure an embedded copy of any individual member of the family.Finally we show that any subset of the n-dimensional hypercube of positive density will contain exponentially many points from some embedded d-dimensional subcube if n is sufficiently large.