The spectrum for overlarge sets of directed triple systems

A directed triple system of order v, denoted by DTS(v, ⋋), is a pair (X, )where X is a v-set and is a collection of transitive triples on X such that every ordered pair of X belongs to ⋋ triples of . An overlarge set of disjoint DTS(v, ⋋) denoted by OLDTS(v, ⋋), is a collection {(Y\{y}, )} _i , such that Y is a (v+1)-set, each (Y\{y}, ) is a DTS (v, ⋋) and all ’s form a partition of all transitive triples of Y. In this paper, we shall discuss the existence problem of OLDTS(v, ⋋) and give the following conclusion: there exists an OLDTS(v, ⋋) if and only if either ⋋ = 1 and v≡ 0,1 (mod 3), or ⋋ = 3 and v ≠ 2. This work was partially supported by the National Natural Science Foundation of China(Grant No. 10671055), Tianyuan Mathematics Foundation of NSFC (Grant No. 10526032) and the Natural Science Foundation of Universities of Jiangsu Province (Grant No. 05KJB110111)