A general recursive construction for quadruple systems

A Steiner-quadruple system of order υ is an ordered pair (X, Q), where X is a set of cardinality υ, and Q is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. In this paper we show that if there exists a quadruple system of order V with a subsystem of order υ, then there exists a quadruple system of order 3V - 2υ with subsystems of orders V and υ. Hanani has given a proof of this result for υ = 1, and in a previous paper, the author has proved the case when V ≡ 2υ(mod 6). The construction given here proves all remaining cases, and has many applications to other existence problems for 3-designs.