  We study random subcube intersection graphs, that is, graphs obtained by
selecting a random collection of subcubes of a fixed hypercube $Q_d$ to serve
as the vertices of the graph, and setting an edge between a pair of subcubes if
their intersection is non-empty. Our motivation for considering such graphs is
to model `random compatibility' between vertices in a large network. For both
of the models considered in this paper, we determine the thresholds for
covering the underlying hypercube $Q_d$ and for the appearance of s-cliques. In
addition we pose some open problems.
