Partitions of sets of designs on seven, eight and nine points

Abstract

We consider the sets of all possible Steiner triple systems (STS) which can be defined on a 7-set or an 8-set, the sets of all possible Steiner quadruple systems (SQS) which can be defined on an 8-set or a 9-set, and the set of all possible Steiner triple systems, on 9 points each, which can be defined on a 9-set. By considering the large and overlarge sets of these designs, we derive various strongly regular graphs and balanced or partially balanced designs. We show connections between the overlarge sets of SQS(8), the STS(9) and the resolutions of the set of all ( 9 3 ) triples chosen from a 9-set into 28 parallel classes of three pairwise disjoint triples, with no two parallel classes orthogonal. Finally we show that the set of all 840 distinct STS(9)'s which can be defined on a given 9-set can be partitioned into 120 large sets of STS(9).