Purely tetrahedral quadruple systems

摘要

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 n-element 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).