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[formula]. 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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics