On universal graphs without instances of CH
type: Item
instance of: scholarly article
title: On universal graphs without instances of CH
zbMATH Open document ID: 0551.03032
DOI: 10.1016/0168-0072(84)90042-3
author: S. Shelah
published in: Annals of Pure and Applied Logic
publication date: 1984-01-01T00:00:00Z
review text: The author proves the striking result that a universal graph on \(\omega_ 1\) may exist under \(2^{\aleph_ 0}=\aleph_ 2\). It is assumed that in the ground model there are \(\aleph_ 2\) stationary subsets of \(\omega_ 1\) with pairwise finite intersetions, this can be achieved by Baumgartner's thinning-out method [\textit{J. E. Baumgartner}, Ann. Math. Logic 9, 401-439 (1976; Zbl 0339.04003)]. First force a graph on \(\omega_ 1\) with finite conditions, then in an iteration up to \(\omega_ 2\) the \(\alpha\) th graph will be pushed into the \(\alpha\) th stationary set. The conditions and the iteration are of finite support with some extra stipulations, similar to Shelah's proof of Baumgartner's theorem on \(\aleph_ 1\)-dense sets [\textit{J. E. Baumgartner}, Ordered sets, Proc. NATO Adv. Study Inst., Banff/Can. 1981, 239-277 (1982; Zbl 0506.04003)]. The main part of the proof is checking ccc of \(P_{\alpha}\) by induction on \(\alpha\), along with proving that some well-behaved conditions are dense. Some generalizations are also proved.
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 3877172
zbMATH Keywords: proper forcing
zbMATH Keywords: universal graph
zbMATH Keywords: stationary subsets
zbMATH Keywords: thinning-out
zbMATH Keywords: ccc
Wikidata QID: Q29040357
reviewed by: Péter Komjáth
MaRDI profile type: Publication
full work available at URL: 0168-0072(84)90042-3
OpenAlex ID: W2012014930
cites work: Powers of regular cardinals
cites work: Independence results
cites work: Almost-disjoint sets the dense set problem and the partition calculus
cites work: On universal locally finite groups
DBLP publication ID: journals/apal/Shelah84
description: scientific article; zbMATH DE number 3877172