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

For every v ≡ 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 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, B1) and (S, B2) of order v having a common flower F ∗ such that |(B1 \ F ∗) ∩ (B2 \ F ∗)| = k. 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, B1) and (S, B2) of order v containing the same block b of size 5 such that |(B1 \ b) ∩ (B2 \ b)| = k. The object of this work is to give a complete solution of both problems. Apart from being of interest in their own right, these solutions can be used to obtain Quattrocchi’s results on maximum packings [1] as a corollary (modulo a few exceptions).