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ON WIDE ARONSZAJN TREES IN THE PRESENCE OF MA

Abstract : A wide Aronszajn tree is a tree of size and height ω1 with no uncount- able branches. We prove that under MA(ω1) 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 MA(ω1), every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todorčević from 2007, gives that under MA(ω1) every wide Aronszajn tree embeds into a Lipschitz tree or a coherent tree. We also prove that under MA(ω1) 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 MA(ω1) there are no universal Aronszajn trees.
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Mirna Džamonja, Saharon Shelah. ON WIDE ARONSZAJN TREES IN THE PRESENCE OF MA. The Journal of Symbolic Logic, Association for Symbolic Logic, In press, ⟨10.1017/jsl.2020.42⟩. ⟨hal-02977169⟩

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