The existence spectrum for overlarge sets of pure directed triple systems

A directed triple system of order v, denoted by DTS(v), is a pair (X, \( \mathcal{B} \)) where X is a v-set and \( \mathcal{B} \) is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of \( \mathcal{B} \). A DTS(v) (X, \( \mathcal{A} \)) is called pure and denoted by PDTS(v) if (a, b, c) ∈ \( \mathcal{A} \) implies (c, b, a) ∉ \( \mathcal{A} \). An overlarge set of PDTS(v), denoted by OLPDTS(v), is a collection {(Y {y i }, \( \mathcal{A}_i^j \)): y i ∈ Y, j ∈ Z 3}, where Y is a (v + 1)-set, each (Y {y i }, \( \mathcal{A}_i^j \)) is a PDTS(v) and these \( \mathcal{A}_i s \) form a partition of all transitive triples on Y. In this paper, we shall discuss the existence problem of OLPDTS(v) and give the following conclusion: there exists an OLPDTS(v) if and only if v ≡ 0,1 (mod 3) and v > 3.