Further results on large sets of disjoint group-divisible designs
type: Item
instance of: scholarly article
title: Further results on large sets of disjoint group-divisible designs
zbMATH Open document ID: 0778.05011
DOI: 10.1016/0012-365X(92)90698-F
author: D. Chen
author: C. C. Lindner
author: D. R. Stinson
published in: Discrete Mathematics
publication date: 1993-05-16T00:00:00Z
review text: A group-divisible design \(\text{GDD}(t,k,rs;r\{s\})\) is a triple \((S,\beta,\gamma)\), where \(S\) is an \(rs\)-set, \(\gamma\) is a partition of \(S\) into \(r\) subsets of \(s\) elements each and \(\beta\) is a collection of \(k\)-subsets of \(S\) such that \(| B\cap G|\leq 1\) for all \(B\in\beta\) and \(G\in\gamma\) and such that each \(t\)-subset \(T\) of \(S\), satisfying \(| T\cap G|\leq 1\) for all \(G\in\gamma\), is contained in exactly one element of \(\beta\). The elements of \(S\), \(\beta\) and \(\gamma\) are called points, blocks and groups, respectively. A large set of disjoint group-divisible designs, or \(\text{LGDD}(t,k,rs;r\{s\})\) is a collection \((S,\beta_ j,\gamma)_{j\in J}\) such that \((S,\beta_ j,\gamma)\) is a \(\text{GDD}(t,k,rs;r\{s\})\) for all \(j\in J\) and such that every \(k\)-subset \(B\) of \(S\) satisfying \(| B\cap G|\leq 1\) for all \(G\in\gamma\) is contained in \(\beta_ j\) for exactly one \(j\in J\). Prior to the present paper, several infinite classes of \(\text{LGDD}(2,3,rs;r\{s\})\) where known, but the authors give new constructions, resulting in new infinite classes, especially for \(s=2\). Since the publication of the paper under review, several further results on the problem have been found. An up to date account of the present state of knowledge about \(\text{LGDD}(2,3,rs;r\{s\})\) can be found in a recent paper by the reviewer [J. Comb. Designs 1, 69-94 (1993)], where the existence problem for \(\text{LGDD}(2,3,rs,r\{s\})\) is settled for all pairs \((r,s)\), except those for which \(r\equiv 2\pmod 6\) and \(s\equiv 0\pmod 6\).
reviewed by: Luc Teirlinck
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 166351
zbMATH Keywords: group-divisible design
zbMATH Keywords: large set
MaRDI profile type: Publication
cites work: Q4201562
cites work: Q4014422
cites work: Construction of large sets of pairwise disjoint transitive triple systems
cites work: On large sets of disjoint Steiner triple systems. VI
cites work: Q3829472
cites work: A Combinatorial Approach to Threshold Schemes
cites work: A completion of Lu's determination of the spectrum for large sets of disjoint Steiner triple systems
description: scientific article; zbMATH DE number 166351