  A paradigm that was successfully applied in the study of both pure and
algorithmic problems in graph theory can be colloquially summarized as stating
that "any graph is close to being the disjoint union of expanders". Our goal in
this paper is to show that in several of the instantiations of the above
approach, the quantitative bounds that were obtained are essentially best
possible. These results are obtained as corollaries of a new family of graphs,
which we construct by picking random subgraphs of the hypercube, and analyze
using (simple) arguments from the theory of metric embedding.
