I gave an invited talk at the 14th International Workshop on Set Theory in Luminy in Marseille, October 2017.
Talk Title: Distributive Aronszajn trees
Abstract: It is well-known that that the statement “all $\aleph_1$-Aronszajn trees are special” is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986). Assume GCH and that $\lambda$ is singular cardinal. If there exists a special $\lambda^+$-Aronszajn tree, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
This suggests that following stronger statement:
Conjecture. Assume GCH and that $\lambda$ is a singular cardinal.
If there exists a $\lambda^+$-Aronszajn tree, then there exists one which is $\lambda$-distributive.
The assumption that there exists a $\lambda^+$-Aronszajn tree is a very mild square-like hypothesis (that is, $\square(\lambda^+,\lambda)$). In order to bloom a $\lambda$-distributive tree from it, there is a need for a toolbox, each tool taking an abstract square-like sequence and producing a sequence which is slightly better than the original one. For this, we introduce the monoid of postprocessing functions and study how it acts on the class of abstract square sequences. We establish that, assuming GCH, the monoid contains some very powerful functions. We also prove that the monoid is closed under various mixing operations.
This allows us to prove a theorem which is just one step away from verifying the conjecture:
Theorem 1. Assume GCH and that $\lambda$ is a singular cardinal.
If $\square(\lambda^+,<\lambda)$ holds, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
Another proof, involving a 5-steps chain of applications of postprocessing functions, is of the following theorem.
Theorem 2. Assume GCH. If $\lambda$ is a singular cardinal and $\square(\lambda^+)$ holds, then there exists a $\lambda^+$-Souslin tree which is coherent mod finite.
This is joint work with Ari Brodsky.