Smallest defining sets of directed triple systems

A directed triple system of order vv, DTS(v), is a pair (V,B)(V,B) where VV is a set of vv elements and BB is a collection of ordered triples of distinct elements of VV with the property that every ordered pair of distinct elements of VV occurs in exactly one triple as a subsequence. A set of triples in a DTS(v)DD is a defining set for DD if it occurs in no other DTS(v) on the same set of points. A defining set for DD is a smallest defining set for DD if DD 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 Dnumber of triples in D. We show that for all v≡0,1(mod3)v≡0,1(mod3), v≥3v≥3 there exists a DTS with f≥12, and improve this result for certain residue classes. In particular, we show that for all v≡1(mod18)v≡1(mod18), v≥19v≥19 there exists a DTS with f≥23. We also prove that, for all ϵ>0ϵ>0 and all sufficiently large admissible vv, there exists a DTS(v) with f≥23−ϵ.Results are also obtained for pure, regular and Mendelsohn directed triple systems.