On wide Aronszajn trees in the presence of MA

Mirna Dzamonja, Saharon Shelah

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
12 Downloads (Pure)

Abstract

A wide Aronszajn tree is a tree of size and height with no uncountable branches. We prove that under there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994. We also prove that under, every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todorčević from 2007, gives that under every wide Aronszajn tree embeds into a Lipschitz tree or a coherent tree. We also prove that under there is no wide Aronszajn tree which weakly embeds all Aronszajn trees, improving the result in the first paragraph as well as a result of Todorčević from 2007 who proved that under there are no universal Aronszajn trees.

Original languageEnglish
Pages (from-to)210-223
Number of pages14
JournalJournal of Symbolic Logic
Volume86
Issue number1
Early online date7 Sep 2020
DOIs
Publication statusPublished - Mar 2021

Keywords

  • wide Aronszajn tree
  • Martin Axiom
  • universality

Cite this