The intersection problem for PBD(5*,3)s

Abstract

For every v equivalent to 5 (mod 6) there exists a pairwise balanced design (PBD) of order v with exactly one block of size 5 and the rest of size 3. We will refer to such a PBD as a PBD(5*, 3). A flower in a PBD(5*, 3) is the set of all blocks containing a given point. If (S, B) is a PBD(5*, 3) and F is a flower, we will write F* to indicate that F contains the block of size 5. The intersection problem for PBD(5*, 3)s is the determination of all pairs (v, k) such that there exists a pair of PBD(5*, 3)s (S, B-1) and (S, B-2) of order v containing the same block b of size 5 such that vertical bar(B-1\b) boolean AND (B-2\b)vertical bar = k. The flower intersection problem for PBD(5*, 3)s is the determination of all pairs (v, k) such that there exists a pair of PBD(5*, 3)s (S, B-1) and (S, B-2) of order v having a common flower F* such that vertical bar(B-1\F*) boolean AND (B-2\F*)vertical bar = k. In this paper we give a complete solution of both problems.