3BD Closed Sets And Their Applications

t-wise balanced designs are very important in design theory. For t = 2, much work has been done on pairwise balanced designs. For t = 3, however, not much is known. The 3BD closed sets B3({4}) and B3({4. 6}) were determined by H. Hanani in 1960 and 1963. respectively. Since then there have been no further results on 3BD closed sets, except the fundamental construction for 3BDs by A. Hartman. In this dissertation, two new 3BD closed sets B3({4.5}) and B3({4,5, 6}) are determined, i.e.,B3(14,5}) = {v>0:v ≡ 1,2,4,5,8,10 (mod 12), v≠13}, B3({4, 5, 6}) = {v > 0 : v ≡ 0, 1. 2 (mod 4), v ≠ 9,13}.The above 3BD closed set B3({4,5, 6}) has been used to solve the existence of large sets of disjoint packings on 6k+5 points by the author and collaborators. In this dissertation, the closed sets B3({4.6}) and B3({4, 5,6}) are used to give a new proof for the existence of large sets of disjoint Steiner triple systems LSTS(6A: + 3). The existence of LSTS(v)s has been completely solved by Lu Jiaxi (1983, 1984) and L. Teirlinck (1991). Since their proof is very complicated, it is much desired to find a simple proof. For this purpose, we introduce a new type of design, a partitionable candelabra system (PCS). Finally, we use it and the 3BD closed sets to obtain the new proof, and to provide an approach to prove the existence of LSTS(36k + 13)s.As a by-product of the determination for B3({4, 5}), we also solve a conjecture given by C. Lindner: The necessary condition v ≡ 4 (mod 12) for the existence of a Steiner quadruple system SQS(v) with a subdesign S(2,4, v) is also sufficient.