A Completion of the Spectrum for Large Sets of Disjoint Transitive Triple Systems

In what follows, an ordered pair will always be an ordered pair (x, y), where x # y. A transitive triple is a collection of three ordered pairs of the form (6, Y), (Y, z), (x, z)}, which we will always denote by (x, y, z). A transitive triple system (TTS(u)) is a pair (X, B), where X is a set containing v elements and B is a collection of transitive triples of elements of X such that every ordered pair of elements of X belongs to exactly one transitive triple of B. It is well known that the spectrum for TTS(U) is the set of all v = 0 or 1 (mod 3). It is easy to see that if (X, B) is a TTS(u) then (BI = v(u 1)/3. A large set of pairwise disjoint TTS(v)s is a collection of 3(v-2) pairwise disjoint TTS(v)s. It is denoted by LTTS(o). Up to now, all known results about LTTS(V) are (see [l-3])