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Distributive Aronszajn trees

Volume 245 / 2019

Ari Meir Brodsky, Assaf Rinot Fundamenta Mathematicae 245 (2019), 217-291 MSC: Primary 03E05; Secondary 03E65, 03E35, 05C05. DOI: 10.4064/fm542-4-2018 Published online: 10 January 2019

Abstract

Ben-David and Shelah proved that if $\lambda $ is a singular strong-limit cardinal and $2^\lambda =\lambda ^+$, then $\square ^*_\lambda $ entails the existence of a normal $\lambda $-distributive $\lambda ^+$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square ^*_\lambda $ by $\square (\lambda ^+,{ \lt }\lambda )$.

As $\square (\lambda ^+,{ \lt }\lambda )$ 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, $\square (\kappa )$ entails the existence of a partition of $\kappa $ into $\kappa $ many fat sets. When contrasted with a classical 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.

Authors

  • Ari Meir BrodskyDepartment of Mathematics
    Bar-Ilan University
    Ramat-Gan 5290002, Israel
    and
    Department of Mathematics
    Ariel University
    Ariel 4070000, Israel
    e-mail
  • Assaf RinotDepartment of Mathematics
    Bar-Ilan University
    Ramat-Gan 5290002, Israel
    e-mail

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