Construction of Large Sets of Pairwise Disjoint Transitive Triple Systems

A transitive triple is a collection of three ordered pairs of the form {(a, b), (b, c), (a, c)}, where a, b, c are all distinct. A transitive triple system (TTS) of order v is a pair (S, T) where S is a set containing v elements and T is a collection of transitive triples of elements of S such that every ordered pair of distinct elements of S belongs to exactly one transitive triple of T. For all v ≡ 0 or 1 (mod 3), it is well-known that a TTS exists, and that |T| = v(v − 1)/3. Since there are altogether v(v − 1)(v − 2) transitive triples of elements of S, it is natural to ask whether the collection of all transitive triples can be partitioned into 3(v − 2) pairwise disjoint TTSs, or failing that, to find the largest positive integer D(v) for which D(v) pairwise disjoint TTSs of order v exist. We show that D(3v) ⩾ 6v + D(v), and D(3v + 1) ⩾ 6v + D(v + 1), for v ⩾ 2