Repeated blocks in indecomposable twofold triple systems

A Steiner triple system (STS) (a twofold triple system (TTS)) is a pair (V, B) where V is a v-set, and B is a collection of 3-subsets of V called blocks or triples such that every 2-subset of V is contained in exactly one [exactly two] blocks. The number v is called the order of the STS or TI'S. It is well-known that an STS of order v exists if and only if v 1 or 3 (mod 6), and a T r s of order v exists if and only if v ~ 0 or 1 (rood 3). A TI'S(v) (V, B) whose block-set B can be partitioned into two subsets B1, B2 such that (V, B0, (V, B2) are both STS(v)'s, is called decomposable; otherwise, it is indecomposable. An indecomposable "ITS(v) is known to exist if and only if v -= 0 or 1 (rood 3) and v :/: 3, 7 [2]. If a T r s contains two blocks bl = {x, y, z}, b2 = {x, y, z} that are identical as subsets of V then {x, y, z} is said to be a repeated block; otherwise, a block is called nonrepeated. In a recent paper [5], the author and D. Hotirnan gave a complete answer to the following question: