Embedding Steiner triple systems into Steiner systems \(S(2,4,v)\).
type: Item
instance of: scholarly article
title: Embedding Steiner triple systems into Steiner systems \(S(2,4,v)\).
zbMATH Open document ID: 1033.05016
DOI: 10.1016/S0012-365X(03)00089-X
author: Mariusz Meszka
author: Alexander Rosa
published in: Discrete Mathematics
publication date: 2004-01-26T00:00:00Z
review text: A Steiner system \(S(t,k,v)\) \((V,{\mathcal B})\) is embedded in a Steiner system \(S(t',k',w)\) \((W,{\mathcal C})\) if \(V \subset W\) and \({\mathcal C}| V = {\mathcal B}\). So far the best studied examples of embeddings of Steiner systems are those when \(t = t' = 2\) or \(3\) and \(k = k' = 3\) or \(4\). In this paper the authors concentrate on the question of embeddings of Steiner triple systems into Steiner systems \(S(2,4,v)\) (i.e. \(t = t' = 2\), \(k = 3\) and \(k' = 4\)). They settle the existence of embeddings of the unique \(\text{STS} (7)\) and, with one possible exception, of the unique \(\text{STS} (9)\). In particular they show that an \(\text{STS} (2,4,w)\) containing an \(\text{STS} (7)\) exists if and only if \(w \geq 25\), \(w \equiv 1,4 \pmod{12}\), and that an \(\text{STS} (2,4,w)\) containing an \(\text{STS} (9)\) exists if and only if \(w = 13\) or \(w \geq 28\), \(w \equiv 1,4 \pmod{12}\), except possibly when \(w = 37\). They also obtain bounds for embedding sizes of Steiner triple systems of larger orders. They conclude by mentioning some open problems, some of which relate to outstanding colouring problems.
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 2032862
zbMATH Keywords: Steiner triple systems
zbMATH Keywords: embedding
reviewed by: Peter B. Gibbons
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2032862