A strongly non-Ramsey order type
type: Item
instance of: scholarly article
title: A strongly non-Ramsey order type
zbMATH Open document ID: 0902.03032
DOI: 10.1007/BF01215918
author: Andras Hajnal
author: Péter Komjáth
published in: Combinatorica
publication date: 1998-11-22T00:00:00Z
review text: Given two order types \(\psi\), \(\theta\) and a cardinal \(\kappa\), \(\psi \nrightarrow[\theta]^2_{\kappa}\) asserts the existence of an ordered set \((X,<)\) of order type \(\psi\) and a function \(f:[X^2]\to\kappa\) such that for every subset \(Y\) of \(X\) of order type \(\theta\), the range of \(f\) on \([Y]^2\) is all of \(\kappa\). It is shown that if a Cohen real is added to a model of the Souslin Hypothesis, then in the extension there is an order type \(\theta\) of cardinality \(\aleph_1\) such that \(\psi \nrightarrow [\theta]^2_{\omega}\) for every order type \(\psi\). Moreover, it is consistent that there is an order type \(\theta\) of cardinality \(\aleph_1\) such that \(\psi\nrightarrow [\theta]^2_{\omega_1}\) for every order type \(\psi\). The authors observe that the following statement was shown consistent by \textit{S. Shelah} [Lect. Notes Math. 1401, 167-193 (1989; Zbl 0683.04002)]: Given an order type \(\theta\) and a cardinal \(\kappa\), there is an order type \(\psi\) with the following property: for every ordered set \((X, <)\) of order type \(\psi\) and every \(f:[X]^2\to \kappa\), there is a subset \(Y\) of \(X\) of order type \(\theta\) such that the range of \(f\) on \([Y]^2\) has size at most 2.
reviewed by: Pierre Matet
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 1225499
zbMATH Keywords: order type
zbMATH Keywords: forcing
zbMATH Keywords: square bracket relation
zbMATH Keywords: consistency
zbMATH Keywords: Cohen real
zbMATH Keywords: Souslin hypothesis
MaRDI profile type: Publication
cites work: Combinatorial set theory: Partition relations for cardinals
cites work: Can you take Solovay's inaccessible away?
full work available at URL: https://doi.org/10.1007/bf01215918
OpenAlex ID: W174896486
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description: scientific article; zbMATH DE number 1225499