Distributive Aronszajn trees

Joint work with Ari Meir Brodsky.

Abstract.  Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^{\lambda }={\lambda }^{+}$, then ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$ entails the existence of a $\lambda$-distributive ${\lambda }^{+}$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }^{\ast }$  by $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}({\lambda }^{+},<\lambda )$.

As $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}({\lambda }^{+},<\lambda )$ does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing.

A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for $\kappa$ regular uncountable, $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}(\kappa )$ entails the existence of a partition of $\kappa$ into $\kappa$ many fat  sets. When contrasted with a classic model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that ${\omega }_2$ cannot be split into two fat  sets.

Downloads:

Citation information:

A. M. Brodsky and A. Rinot, Distributive Aronszajn trees. Fund. Math., 245(3): 217-291, 2019.