A theorem and some consistency results in partition calculus
type: Item
instance of: scholarly article
title: A theorem and some consistency results in partition calculus
zbMATH Open document ID: 0625.03029
DOI: 10.1016/0168-0072(87)90015-7
author: S. Shelah
author: Lee J. Stanley
published in: Annals of Pure and Applied Logic
publication date: 1987-01-01T00:00:00Z
review text: For ordinals \(\alpha\), \(\beta\), \(\gamma\) (the order type is concerned, not the cardinality) say \(\alpha \to (\beta,\gamma)^ 2\) iff whenever \(f:[\alpha]^ 2\to 2\) there is an \(x\subset \alpha\) s.t. either x has order type \(\beta\) and \(\forall (a,b)\in [x]^ 2\) \(f(a,b)=0\) or x has order type \(\gamma\) and \(\forall (a,b)\in [x]^ 2\) \(f(a,b)=1.\) It is shown that for regular \(\kappa\) such that \(\forall \lambda <\kappa\), \(\lambda^{\aleph_ 0}<\kappa\) and \(k<\omega\kappa \omega \to (\kappa \omega,k)^ 2\). Various consistency results are given, e.g.: \(ZFC+\omega_ 2\omega \to (\omega_ 2\omega,3)^ 2+\omega_ 3\omega_ 1\nrightarrow (\omega_ 3\omega_ 1,3)^ 2\). These are extensions of the Erdős-Rado and Erdős-Dushnik-Miller theorems.
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 4014699
zbMATH Keywords: partition calculus
zbMATH Keywords: order type
zbMATH Keywords: consistency results
MaRDI profile type: Publication
full work available at URL: 0168-0072(87)90015-7
OpenAlex ID: W2002646945
cites work: Remarks on superatomic Boolean algebras
cites work: Q5635446
cites work: Combinatorial set theory: Partition relations for cardinals
cites work: An ideal game
cites work: A Negative Partition Relation
cites work: Q3852172
cites work: Set theory. An introduction to independence proofs
cites work: Making the supercompactness of \(\nu\) indestructible under \(\nu\)-directed closed forcing
cites work: A note on cardinal exponentiation
cites work: Canonization theorems and applications
cites work: \(\aleph_\omega\) may have a strong partition relation
cites work: S-forcing. I: a ''black-box'' theorem for morasses, with applications to super-Souslin trees
cites work: S-forcing. IIa: Adding diamonds and more applications: Coding sets, Arhangel'skii's problem and \({\mathcal L}[Q_ 1^{<\omega},Q^ 1_ 2]\). (With an appendix by John P. Burgess)
cites work: More consistency results in partition calculus
cites work: Morasses, diamond, and forcing
cites work: Simplified morasses
cites work: Simplified morasses with linear limits
description: scientific article; zbMATH DE number 4014699