Smallest defining sets of directed triple systems

A directed triple system of order v , DTS ( v ) , is a pair ( V , B ) where V is a set of v elements and B is a collection of ordered triples of distinct elements of V with the property that every ordered pair of distinct elements of V occurs in exactly one triple as a subsequence. A set of triples in a DTS ( v ) D is a defining set for D if it occurs in no other DTS ( v ) on the same set of points. A defining set for D is a smallest defining set for D if D has no defining set of smaller cardinality. In this paper we are interested in the quantity f = number of triples in a smallest defining set for  D number of triples in  D . We show that for all v ≡ 0 , 1 ( mod 3 ) , v ≥ 3 there exists a DTS with f ≥ 1 2 , and improve this result for certain residue classes. In particular, we show that for all v ≡ 1 ( mod 18 ) , v ≥ 19 there exists a DTS with f ≥ 2 3 . We also prove that, for all ϵ > 0 and all sufficiently large admissible v , there exists a DTS ( v ) with f ≥ 2 3 − ϵ . s are also obtained for pure, regular and Mendelsohn directed triple systems.