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arXiv · 2002.02396

On wide Aronszajn trees in the presence of MA

Mirna Džamonja·Saharon Shelah
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Abstract

A wide Aronszajn tree is a tree of size and height $ω_1$ with no uncountable branches. We prove that under $MA(ω_1)$ 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 the same assumption there is no universal Aronszajn tree, improving a result of Todorčevi{ć} from 2007 who proved the same under the assumption of BPFA for posets of size $\aleph_1$. Finally, we prove that under $MA(ω_1)$, every wide Aronszajn tree weakly embeds in an Aronszajn tree.

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Mirna Džamonja, Saharon Shelah. 2020-10-24. On wide Aronszajn trees in the presence of MA. https://doi.org/10.1017/jsl.2020.42

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