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 ≥ 23 . We also prove that, for all > 0 and all sufficiently large admissible v, there exists a DTS(v) with f ≥ 23 − . Results are also obtained for pure, regular and Mendelsohn directed triple systems.