An extremal theorem in the hypercube
Abstract
The hypercube Qn is the graph whose vertex set is 0,1n 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.
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