Pairwise balanced designs whose block size set contains seven and thirteen

Abstract

Abstract:In this paper, we investigate the PBD-closure of sets K with {7, 13} ⊆ K ⊆ {7, 13, 19, 25, 31, 37, 43}. In particular, we show that v ≡ 1 mod 6, v ≥ 98689 implies v ∈ B({7, 13}). As an intermediate result, many new 13-GDDs of type 13 q and resolvable BIBD with block size 6 or 12 are also constructed. Furthermore, we show some elements to be not essential in a Wilson basis for the PBD-closed set {v : v ≡ 1 mod 6, v ≥ 7}.