The μ-way intersection problem for cubes

A collection of Il 3-cube decompositions of a graph G is said to have intersection size t if there is a set S, with 181 = t, of 3-cubes which is contained in every decomposition, and no 3-cube which is not in S occurs in more than one decomposition. We determine all integers n, t and Il, with Il ::; 12, for which there exists a collection of Il 3-cube decompositions of Kn with intersection size t. Also, we determine all integers m, n, t and Il, with Il ::; 8, for which there exists a collection of Il 3-cube decompositions of Km,n with intersection size t.