Construction of Large Sets of Pairwise Disjoint Transitive Triple Systems

Abstract

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 (3 v ) ⩾ 6 v + D ( v ), and D (3 v + 1) ⩾ 6 v + D ( v + 1), for v ⩾ 2.