Mirna Džamonja (CNRS – Université de Paris)

On the universality problem for $\aleph_2$-Aronszajn and wide $\aleph_2$-Aronszajn trees

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We report on a joint work in progress with Rahman Mohammadpour in which we study the problem of the possible existence of a universal tree under weak embeddings in the classes of $\aleph_2$-Aronszajn and wide $\aleph_2$-Aronszajn trees. This problem is more complex than previously thought, in particular it seems not to be resolved under ShFA + CH using the technology of weakly Lipshitz trees. We show that under CH, for a given $\aleph_2$-Aronszajn tree $T$ without a weak ascent path, there is an $\aleph_2$-cc countably closed forcing forcing which specialises $T$ and adds an $\aleph_2$-Aronszajn tree which does not embed into $T$. One cannot however apply the ShFA to this forcing.

Further, we construct a model à la Laver-Shelah in which there are $\aleph_2$-Aronszajn trees, but none is universal. Work in progress is to obtain an analogue for universal wide $\aleph_2$-Aronszajn trees. We also comment on some negative ZFC results in the case that the embeddings are assumed to have a strong preservation property.