Complete subgraphs of infinite multipartite graphs and antichains in partially ordered sets
type: Item
instance of: scholarly article
title: Complete subgraphs of infinite multipartite graphs and antichains in partially ordered sets
zbMATH Open document ID: 0655.05002
DOI: 10.1016/0012-365X(86)90069-5
published in: Discrete Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The authors investigate a conjecture of E. C. Milner and Sauer: ``If the cofinality of a partially ordered set (P,\(\leq)\) is a singular cardinal \(\lambda\), then P contains an antichain of size cf(\(\lambda)\)''. (The cofinality of (P,\(\leq)\) is the least cardinality of a cofinal subset. The word ``antichain'' has the nonstandard meaning of a set of pairwise incomparable elements.) The conjecture was inspired by a result of \textit{M. Pouzet} [``Parties cofinals des ordres partiels ne contenant pas d'antichaines infinies'', J. Lond. Math. Soc., 2nd Ser. (to appear)] concluding under the same hypothesis that P contains an infinite antichain. The authors prove the conjecture under certain additional assumptions on the cardinal \(\lambda\). They also prove that, if the conjecture fails for some cardinal \(\lambda\), then there must exist some \(\lambda '<\lambda\) and a set S of subsets of \(\lambda\) ', ordered by inclusion, which provides a counterexample.
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4068875
zbMATH Keywords: cofinality
zbMATH Keywords: partially ordered set
zbMATH Keywords: antichain
author: Andras Hajnal
author: Norbert Sauer
MaRDI profile type: Publication
cites work: The Cofinality of a Partially Ordered Set
cites work: Partition relations for cardinal numbers
cites work: Q4088825
full work available at URL: 0012-365x(86)90069-5
OpenAlex ID: W2065535874
description: scientific article; zbMATH DE number 4068875