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On wide Aronszajn trees in the presence of MA

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  • Mirna Dzamonja
  • Saharon Shelah

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Abstract

A wide Aronszajn tree is a tree of size and height $\omega_1$ with no uncountable branches. We prove that under $MA(\omega_1)$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and
V\"a\"an\"anen from 1994.

We also prove that under $MA(\omega_1)$, every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todor{\v c}evi{\'c} from 2007, gives that under $MA(\omega_1)$ every wide Aronszajn tree embeds into a Lipschitz tree or a coherent tree. We also prove that under $MA(\omega_1)$ 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 Todor{\v c}evi{\'c} from 2007 who proved that under $MA(\omega_1)$ there are no universal Aronszajn trees.

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Original languageEnglish
Number of pages18
JournalJournal of Symbolic Logic
Early online date7 Sep 2020
DOIs
Publication statusPublished - Sep 2020
Peer-reviewedYes

Keywords

    Research areas

  • wide Aronszajn tree, Martin Axiom, universality

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