An Extremal Theorem in the Hypercube

The hypercube $Q_n$ 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 $\textrm{ex}(Q_n, H)$ be the maximum number of edges in a subgraph of $Q_n$ which does not contain a copy of $H$. We find a wide class of subgraphs $H$, including all previously known examples, for which $\textrm{ex}(Q_n, H) = o(e(Q_n))$. In particular, our method gives a unified approach to proving that $\textrm{ex}(Q_n, C_{2t}) = o(e(Q_n))$ for all $t \geq 4$ other than $5$.