Purely tetrahedral quadruple systems  

作　　者：JI Lijun 

机构地区：[1]Department of Mathematics, Suzhou University, Suzhou 215006, China

出　　处：《Science China Mathematics》2006年第10期1327-1340,共14页中国科学：数学（英文版）

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).

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),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).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).

关 键 词：large set  t-wise BALANCED design  MENDELSOHN TRIPLE system  quadruple system. 

分 类 号：N[自然科学总论]