# On wide Aronszajn trees in the presence of MA

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### Authors

• Mirna Dzamonja
• Saharon Shelah

### 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.

### Details

Original language English 18 Journal of Symbolic Logic 7 Sep 2020 https://doi.org/10.1017/jsl.2020.42 Published - Sep 2020 Yes

### Research areas

• wide Aronszajn tree, Martin Axiom, universality

### Bibliographic note

The accepted version not put in the journal form

ID: 182249886

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