Two negative partition relations
type: Item
instance of: scholarly article
title: Two negative partition relations
zbMATH Open document ID: 0639.03053
DOI: 10.1007/BF01849028
author: J. Takahashi
published in: Periodica Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: Hajnal has shown that the continuum hypothesis implies \(\omega_ 1\cdot \omega \nrightarrow (\omega_ 1\cdot \omega,3)^ 2\) and \(\omega_ 1\cdot \omega_ 1\nrightarrow (\omega_ 1\cdot \omega_ 1,3)^ 2\). Call the following set-theoretic hypotheses \(\Gamma\) and \(\Delta\) respectively: (a) There exists a family \({\mathcal A}\subseteq [\omega_ 1]^{\omega}\) such that card(\({\mathcal A})=\omega_ 1\) and \(\forall X\in [\omega_ 1]^{\omega_ 1}:\exists A\in {\mathcal A}:\) \(A\subseteq X\); (b) There exists a family \({\mathcal G}\subseteq^{\omega}\omega\) such that card(\({\mathcal G})=\omega_ 1\) and \(\forall f\in^{\omega}\omega:\exists g\in {\mathcal G}:\forall n<\omega:\) \(f(n)<g(n)\). The conjunction of \(\Gamma\) and \(\Delta\) is known to be weaker than the continuum hypothesis. In this paper it is shown that \(\omega_ 1\cdot \omega \nrightarrow (\omega_ 1\cdot \omega,3)^ 2\) follows from the conjunction of \(\Gamma\) and \(\Delta\), and \(\omega_ 1\cdot \omega_ 1\nrightarrow (\omega_ 1\cdot \omega_ 1,3)^ 2\) from \(\Gamma\).
reviewed by: J. Takahashi
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 4041905
zbMATH Keywords: partition relations
zbMATH Keywords: weakenings of the continuum hypothesis
MaRDI profile type: Publication
cites work: Almost-disjoint sets the dense set problem and the partition calculus
cites work: A Negative Partition Relation
full work available at URL: https://doi.org/10.1007/bf01849028
OpenAlex ID: W1986619830
description: scientific article; zbMATH DE number 4041905