An Existence Theory for Pairwise Balanced Designs I. Composition Theorems and Morphisms

One of the most central problems of modern combinatorial theory is the determination of those parameter triples (v, k, X) for which there exist (0, k, h)-BIBD’s (balanced incomplete block designs). The methods which have been put forth to construct BIBD’s divide roughly into two classes: direct constructions where a BIBD is obtained from an algebraic structure (often a design is constructed from its automorphism group), and recursive or composition methods where a BIBD is built up by purely combinatorial means from another design or an assortment of “smaller” designs (see [6]). It is the second class with which we deal here. One of the first instances of a composition theorem was presented by E. H. Moore [l l] in 1893 in connection with Steiner triple systems, i.e., (0, 3, I)-BIBD’s. Several methods were expounded by Bose and Shrikhande [l] and composition techniques were instrumental in their remarkable work (with E. T. Parker) on orthogonal Latin squares [2, 31. Significant contributions have been made by H. Hanani [7, 8, lo] in his work on BIBD’s with k = 3,4, 5 and composition methods are used by Ray-Chaudhuri and Wilson [12] in connection with Kirkman designs. An attempt is made here to unify some of these various constructions and to present them in a more general, common setting. The general theorems presented here will be illustrated by examples and will be applied in the second part of this article, “An Existence Theory for Pairwise Balanced Designs, II. The Structure of PBD-Closed Sets and the Existence Conjectures” [15], to a conjecture on the existence of BIBD’s [6, p. 2381. For simplicity of exposition, we consider only designs with X = 1. We