Purely tetrahedral quadruple systems

作     者：JI Lijun 

作者机构：Department of Mathematics Suzhou University Suzhou 215006 China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2006年第49卷第10期

页      面：1327-1340页

核心收录：

学科分类：07[理学] 08[工学] 

基　　金：This work was partially supported by the Tianyuan Mathematics Foundation of NSFC(Grant No.10526032) the Natural Science Foundation of Universities of Jiangsu Province(Grant No.05KJB110111) 

主　　题：large set, t-wise balanced design, Mendelsohn triple system, quadruple system. 

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 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),n4,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 n4, 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),n3, or n≡0,4 (mod 12).