Intersecting Designs

We prove the intersection conjecture for designs: For any complete graph Kr there is a finite set of positive integers M(r) such that for every n > n0(r), if Kn has a Kr-decomposition (namely a 2-(n, r, 1) design exists) then there are two Kr-decompositions of Kn having exactly q copies of Kr in common for every q belonging to the set {0, 1, . . . , ( n 2 ) / ( r 2 ) }\{ ( n 2 ) / ( r 2 ) −m | m ∈M(r)}. In fact, this result is a special case of a much more general result, which determines the existence of k distinct Kr-decompositions of Kn which have q elements in common, and all other elements of any two of the decompositions share at most one edge in common.