A diamond example of an ordinal graph with no infinite paths
type: Item
instance of: scholarly article
title: A diamond example of an ordinal graph with no infinite paths
zbMATH Open document ID: 0703.03028
DOI: 10.1016/0168-0072(90)90013-R
author: James E. Baumgartner
author: Jean A. Larson
published in: Annals of Pure and Applied Logic
publication date: 1990-01-01T00:00:00Z
review text: Given ordinals \(\alpha\) and \(\beta\), \(\alpha\to (\beta\), infinite \(path)^ 2\) means that every graph on \(\alpha\) either contains an independent (edge-free) set of order type \(\beta\) or an infinite path. It is easy to see that \(\alpha +n+1\nrightarrow (\alpha +1\), infinite \(path)^ 2\) whenever \(\alpha >0\) and \(n<\omega\). \textit{P. Erdős}, \textit{A. Hajnal} and \textit{E. C. Milner} [Combinat. Theory Appl., Colloq. Math. Soc. János Bolyai 4, 327-363 (1970; Zbl 0215.329)] have shown that \(\alpha\to (\alpha\), infinite \(path)^ 2\) for all limit ordinals \(\alpha <\omega_ 1^{\omega +2}\). This positive result has been extended by the second author [in the paper reviewed below (Zbl 0703.03029)] to all limit ordinals \(\alpha <2^{\aleph_ 0}\) under the assumption of Martin's Axiom. The authors show that if Jensen's principle \(\diamondsuit\) holds, then \(\alpha \nrightarrow (\omega_ 1^{\omega +2}\), infinite \(path)^ 2\) for all \(\alpha <\omega_ 2\).
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 4152364
zbMATH Keywords: diamond
zbMATH Keywords: ordinal graph
zbMATH Keywords: tree
zbMATH Keywords: infinite path
MaRDI profile type: Publication
cites work: Q3222179
cites work: Q3852172
cites work: An Independence Result for Pinning for Ordinals
cites work: Martin's axiom and ordinal graphs: Large independent sets or infinite paths
cites work: Teilmengen von Mengen mit Relationen
full work available at URL: 0168-0072(90)90013-r
OpenAlex ID: W2070331185
description: scientific article; zbMATH DE number 4152364