Embeddings on \(S(2,4,v)\)
type: Item
instance of: scholarly article
title: Embeddings on \(S(2,4,v)\)
zbMATH Open document ID: 0688.05010
DOI: 10.1016/S0195-6698(89)80050-2
published in: European Journal of Combinatorics
publication date: 1989-01-01T00:00:00Z
review text: A Steiner system \(S(2,k,v)\) is a pair \((X,\mathcal A)\) where \(X\) is a \(n\)-set and \(\mathcal A\) is a collection of some \(k\)-subsets (called blocks) of \(X\) such that any two distinct elements of \(X\) are contained in exactly one block in \(\mathcal A\). It has been proved that an \(S(2,4,v)\) exists iff \(v\equiv 1\) or \(4\mod{12}\). Any positive integer satisfying this congruence is called admissible. If \((X,\mathcal A)\) and \((Y,\mathcal B)\) are two systems such that \(X\subseteq Y\) and \({\mathcal A}\subseteq {\mathcal B}\), then \((X,\mathcal A)\) is said to be embedded in \((Y,\mathcal B)\). The necessary condition for an \(S(2,k,u)\) to be embedded in an \(S(2,k,v)\) is that \(n\geq (k-1)m+1\). Several authors have investigated the sufficiency [see \textit{J. Doyen} and \textit{R. M. Wilson}, Discrete Math. 5, 229--239 (1973; Zbl 0263.05017), \textit{A. E. Brouwer} and \textit{H. Lenz}, Contribution to geometry, Proc. Symp., Siegen 1978, 383--389 (1979; Zbl 0425.51006), Eur. J. Comb. 2, 323--330 (1981; Zbl 0477.05016)]. It is proved in this paper that if \(u\) and \(v\) are admissible integers and \(u>85\), \(u\neq 133\), then an \(S(2,4,u)\) can be embedded in some \(S(2,4,v)\) iff \(v\geq 3u+1\). Recently, the sufficiency for the remaining cases when \(u\leq 85\) or \(u=133\) has also been obtained by \textit{R. Rees} and \textit{D. R. Stinson} in their paper ``On the existence of incomplete designs of block size four having one hole'' in Utilitas Math. 35, 119--152 (1989; Zbl 0678.05009).
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 4127215
zbMATH Keywords: embedding
zbMATH Keywords: incomplete array
zbMATH Keywords: Steiner system
author: Ruizhong Wei
author: Li Zhu
MaRDI profile type: Publication
cites work: Q3686719
cites work: Optimal packings of \(K_4\)'s into a \(K_n\)
cites work: Q4190650
cites work: Subspaces of linear spaces of line size 4
cites work: More mutually orthogonal latin squares
cites work: Q4769064
cites work: Q5592263
cites work: Balanced incomplete block designs and related designs
cites work: Existence of orthogonal Latin squares with aligned subsquares
cites work: Q3728004
cites work: Q3794192
cites work: Concerning the number of mutually orthogonal latin squares
cites work: Q4071247
cites work: Q3309839
cites work: Four pairwise orthogonal Latin squares of order 24
description: scientific article; zbMATH DE number 4127215