Preprint in the style of Combinatorics , Probability and

Let Gp be a random graph on 2 d vertices where edges are selected independently with a xed probability p > 1 4 , and let H be the d-dimensional hypercube Q d. We answer a question of Bollobb as by showing that, as d ! 1, Gp almost surely has a spanning subgraph isomorphic to H. In fact we prove a result which can be applied to many other graphs, improving also previous results for the lattice, i.e., the 2-dimensional square grid. The proof uses the second moment method|writing X for the number of subgraphs of G isomorphic to H, where G is a suitable random graph, we expand the variance of X as a sum over all subgraphs of H itself. As the subgraphs of H may be quite complicated, most of the work is in estimating the various terms of this sum.