A new existence proof for large sets of disjoint Steiner triple systems

A Steiner triple system of order v (briefly STS ( v ) ) consists of a v -element set X and a collection of 3-element subsets of X, called blocks, such that every pair of distinct points in X is contained in a unique block. A large set of disjoint STS ( v ) (briefly LSTS ( v ) ) is a partition of all 3-subsets (triples) of X into v - 2 STS ( v ) . In 1983–1984, Lu Jiaxi first proved that there exists an LSTS ( v ) for any v ≡ 1 or 3 ( mod 6 ) with six possible exceptions and a definite exception v = 7 . In 1989, Teirlinck solved the existence of LSTS ( v ) for the remaining six orders. Since their proof is very complicated, it is much desired to find a simple proof. For this purpose, we give a new proof which is mainly based on the 3-wise balanced designs and partitionable candelabra systems.