Subdesigns in Steiner quadruple systems

Abstract

A Steiner quadruple system of order v, denoted SQS( v), is a pair ( X, B ), where X is a set of cardinality v, and B is a set of 4-subsets of C (called blocks), with the property that any 3-subset of X is contained in a unique block. If ( X, B ) is an SQS( v) and ( Y, C) is an SQS( w) with Y ⊆ X and C ⊆ B , we say that ( Y, C) is a subdesign of ( X, B ). Hanani has shown that an SQS( v) exists for all v ≡ 2 or 4 (mod 6) and when v ∈ {0, 1}; such integers v are said to be admissible. A necessary condition for the existence of an SQS( v) with a subdesign of order w is that v = w or v ⩾ 2 w. In this paper we show the existence of an explicitly computable constant k (independent of w) such that for all admissible v and all admissible w with v ⩾ kw there exists an SQS( v) containing a subdesign of order w. We also show that for any sufficiently large w we can take k = 12.54. To establish these results we introduce several new constructions for SQS, and we also consider the subdesign problem for related classes of designs.