An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X,B), where X is an nelement set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X, B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n≡2,4(mod 6),n＞4,or n≡1,5(mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n≡2,4 (mod 6) and n＞4, or n≡1, 5 (mod 12).Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n≡1,3(mod 6),n＞3, or n≡0,4 (mod 12).
A directed triple system of order v, denoted by DTS(v, λ), is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint DTS(v, λ), denoted by OLDTS(v, λ), is a collection {(Y\{y}, Ai)}i,such that Y is a (v + 1)-set, each (Y\{y}, Ai) is a DTS(v, λ) and all Ai's form a partition of all transitive triples of Y. In this paper, we shall discuss the existence problem of OLDTS(v, λ) and give the following conclusion: there exists an OLDTS(v, λ) if and only if either λ = 1 and v = 0, 1 (mod 3), or λ = 3 and v≠2.
Zi-hong TIAN~(1+) Li-jun JI~2 ~1 Institute of Mathematics,Hebei Normal University,Shijiazhuang 050016,China
An LPDTS(υ) is a collection of 3(υ - 2) disjoint pure directed triple systems on the same set of υ elements. It is showed in Tian's doctoral thesis that there exists an LPDTS(υ) for υ≡ 0,4 (mod 6), υ≥ 4. In this paper, we establish the existence of an LPDTS(υ) for υ≡ 1, 3 (mod 6), υ＞ 3. Thus the spectrum for LPDTS(υ) is completely determined to be the set {υ:υ≡0, 1 (mod 3),υ≥4}.
ZHOU Junling, CHANG Yanxun & Jl Lijun Institute of Mathematics, Beijing Jiaotong University, Beijing 100044, China
