On Large Sets of Disjoint Steiner Triple Systems II

A Steiner system S(t, k, V) is a pair (S, /3), where S is a v-set and p is a collection of k-subsets of S called hocks, such that a t-subset of S occurs in exactly one block of p. In particular, an S(2, 3, V) is called a Steiner triple system of order v (briefly STS(v)). It is well known that there is an STS(v) if and only if v E 1 or 3 (mod 6). Two STSs, (S, /3,) and (S, &), are said to be disjoint provided p, n/3, = 0. We denote by D(v) the maximum number of pairwise disjoint STSs of order v, and it is easy to show that D(v) < v 2 for v > 1. If D(v) = v 2, then this means that the set of all triples out of a set of size v can be partitioned into v 2 subsets so that each of them forms an STS(v). We call any set of v 2 pairwise disjoint STS(v) a large set of disjoint STS(v) and denote it by LTS(v). The purpose of this paper is to prove: