The \(\mu\)-way intersection problem for cubes (Q2712514)

A great deal of work has been done in the recent past on the intersection problem for combinatorial designs. The question addressed in this respect is: given two designs based on the same underlying set of elements, how many blocks may they have in common? The intersection problem has been considered for many classes of designs, including Steiner triple systems, \(m\)-cycle systems, and Steiner quadruple systems. For a fine survey on the intersection problem, the reader is directed to \textit{E. J. Billington} [J. Comb. Des. 1, No. 6, 435-452 (1993; Zbl 0885.05044)] and the references therein.NEWLINENEWLINENEWLINEThere is no reason to restrict one's attention to the number of blocks that a pair of designs have in common; one may consider the intersection problem for a collection of \(\mu\) designs (defined on the same underlying set). Indeed, this problem has already been considered for \(m\)-cycle systems and, in the case \(\mu= 3\), for Steiner triple systems and Latin squares.NEWLINENEWLINENEWLINEIn this paper the authors consider the \(\mu\)-way intersection problem for decompositions of the complete graph into cubes of dimension three. A collection of \(\mu\) 3-cube decompositions of a graph \(G\) is said to have intersection size \(t\) if there is a set \(S\) of 3-cubes with \(|S|= t\) which is contained in every decomposition, and no 3-cube which is not in \(S\) occurs in more than one decomposition. The authors determine all integers \(n\), \(t\), and \(\mu\), with \(\mu\leq 12\), for which there exists a collection of \(\mu\) 3-cube decompositions of \(K_n\) with intersection size \(t\). They also determine all integers \(m\), \(n\), \(t\), \(\mu\), with \(\mu\leq 8\), for which there exists a collection of \(\mu\) 3-cube decompositions of \(K_{m,n}\) with intersection size \(t\).