单纯Hybrid三元系大集的三倍构作

Tripling Construction for Large Sets of Hybrid Triple Systems

一个v阶Hybrid三元系,记作HTS（v）,是一个对子（X,B）,其中X是v元集,B是X中循环三元组和可迁三元组的集合（称作区组）,满足X的每个有序对都恰包含于B中一个区组。设（X,B）是一个没有重复区组的HTS（v）,如果区组集{（x,y,z）,（z,y,x）,（z,x,y）,（y,x,z）,（y,z,x）,（x,z,y）,〈x,y,z〉,〈z,y,x〉}中有一个三元组包含在B中,必有区组集中其它三元组都不包含在B中,则称（X,B）是单纯的,记为PHTS（v）。不相交PHTS（v）大集,记为LPHTS（v）,是一个集合{（X,Bi）}i,其中每个（X,Bi）都是PHTS（v）,并且∪iBi构成了X中所有循环三元组和可迁三元组的一个划分。给出了LPHTS（v）的一种三倍构造方法,得到了其存在的两个无穷类：对于非负整数m,存在LPHTS（3·3^m＋1）和LPHTS（5·3^m＋1）。

An hybrid triple system of order v,denoted by HTS（v）,is a pair（X,B）where X is a v-set and B is a collection of cyclic triples and transitive triples on X,called blocks,such that every ordered pair of X belongs to exactly one block of B.An HTS（v）without repeated blocks is called pure and denoted by PHTS（v）if one element of the block set{（x,y,z）,（z,y,x）,（z,x,y）,y,x,z）,（y,z,x）,（x,z,y）,x,y,z,z,y,x} is contained in B then the others will not be contained in B.A large set of disjoint PHTS（v）s,denoted by LPHTS（v）,is a collection of {（X,Bi）}i where each（X,Bi）is a PHTS（v）and ∪iBi is a partition of all cyclic and transitive triples on X.A tripling construction for LPHTS（v）is given,and obtained two infinite families for the existence of LPHTS（v）：for any nonnegative integers m,there exists an LPHTS（3·3^m＋1）and an LPHTS（5·3^m＋1）.