Intersecting designs

We prove the intersection conjecture for designs: For any complete graph K- r there is a finite set of positive integers M(r) such that for every n > n (0)(r), if K-n has a K-r -decomposition (namely a 2-(n,r,1) design exists) then there are two K-r-decompositions of K-n, having exactly q copies of K- r in common for every q belonging to the set {0,1, ..., ((2) (n))/((n)(2))} \{( (n)(2))/((n)(2))-m \ m is an element of M(r)}. In fact, this result is a special case of a much more general result, which determines the existenc e of k distinct K-r-decompositions of K-n which have q elements in common, and all other elements of any two of the decompositions share at most one e dge in common. (C) 2000 Academic Press.