On wide Aronszajn trees in the presence of MA

Dzamonja, Mirna and Shelah, Saharon (2020) On wide Aronszajn trees in the presence of MA. Journal of Symbolic Logic. ISSN 0022-4812 (In Press)

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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.

Item Type: Article
Additional Information: The accepted version not put in the journal form
Uncontrolled Keywords: wide aronszajn tree,martin axiom,universality,mathematics(all) ,/dk/atira/pure/subjectarea/asjc/2600
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: LivePure Connector
Date Deposited: 04 Jun 2020 00:07
Last Modified: 02 Jul 2020 00:01
URI: https://ueaeprints.uea.ac.uk/id/eprint/75460
DOI:

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