Large sets of extended directed triple systems with odd orders
type: Item
instance of: scholarly article
title: Large sets of extended directed triple systems with odd orders
zbMATH Open document ID: 1230.05059
author: Yuanyuan Liu
author: Qingde Kang
published in: Discrete Mathematics
publication date: 2010-11-19T00:00:00Z
review text: This paper is concerned with the problem of the existence of large sets of extended directed triple systems with odd orders. Let \(X\) be a finite set with \(v\) elements, and let \(x\), \(y\), \(x\) \((x\neq y\neq z)\) be elements of \(X\). A triple \([x,y,z]\) or (cyclic triple \(\langle x,y,z\rangle\), or transitive triple \((x,y,z)\)) on \(X\) is a set of three unordered pairs \([x,y]\), \([y,z]\), \([z,x]\) (or ordered pairs \((x,y)\), \((y,z)\), \((z,x)\), or ordered pairs \((x,y)\), \((y,z)\), \((x,z)\)) of \(X\). For each of these three types of triples, the corresponding extended triples and their large sets are introduced. An extended Steiner (or Mendelshohn, or directed) triple systems (denoted by \(\text{EMTS}(v)\), or \(\text{EDTS}(v)\)) is a pair \((X,A)\), \(A\) being a collection of extended triples (or cyclic triples, or transitive triples) on \(X\), called blocks, such that each extended unordered (or ordered) pair of \(X\) is contained in exactly one block of \(A\). Corresponding to each extended system, its large set (denoted by \(\text{LEMTS}(v)\), or \(\text{LEMTS}(v)\), or \(\text{LEDTS}(v)\)) is introduced which is a collection \(\langle(X, A_k)\rangle_k\), where \(A_k\) form a partition of all extended triples on \(X\) and each other \((X,A_k)\) is an \(\text{ESTS}(v)\), or \(\text{EMTS}(v)\), or \(\text{EDTS}(v)\). The authors here discuss the existence problem of \(\text{LEDTS}(v)\) when \(v\) is odd. The following are the two main results: Result 1. There exists an \(\text{LEDTS}(v)\) for any odd \(v\) except possible \(v=95\), \(143\), \(167\), \(203\), \(215\). Result 2. There exists an \(\text{LEDTS}(v)\) for any integer \(v\neq 4\) except possible \(v=95\), \(143\), \(167\), \(203\), \(215\).
reviewed by: D. V. Chopra
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 5817555
zbMATH Keywords: extended triple
zbMATH Keywords: extended triple system
zbMATH Keywords: large set
MaRDI profile type: Publication
full work available at URL: j.disc.2010.09.015
OpenAlex ID: W4237433681
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DOI: 10.1016/J.DISC.2010.09.015
description: scientific article; zbMATH DE number 5817555