One-Factorizations and Triple Systems

Abstract

Among the most important combinatorial designs are balanced incomplete block designs. A balanced incomplete block design with parameters (v, b, r, k, λ) is a way of choosing b subsets of size k from a v-set so that any element belongs to precisely r of the sets and any two elements commonly belong to λ of them. The k-sets are called blocks. It is easy to see that these parameters are not independent — in fact, r(k — 1) = λ(v — 1) and bk = vr — and that the constancy of r is implied by the constancy of k and λ.