An Extremal Theorem in the Hypercube

The hypercube Qn is the graph whose vertex set is {0, 1}n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Qn,H) be the maximum number of edges in a subgraph of Qn which does not contain a copy of H. We find a wide class of subgraphs H, including all previously known examples, for which ex(Qn,H) = o(e(Qn)). In particular, our method gives a unified approach to proving that ex(Qn, C2t) = o(e(Qn)) for all t > 4 other than 5.