Distributive Aronszajn trees

Ari Meir Brodsky, Assaf Rinot

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


Ben-David and Shelah proved that if λ is a singular strong-limit cardinal and 2 λ = λ + , then λ entails the existence of a normal λ-distributive λ + -Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis λ by (λ + ,<λ). As (λ + ,<λ) 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 κ regular uncountable, (κ) entails the existence of a partition of κ into κ 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 ω2 cannot be split into two fat sets.

Original languageEnglish
Pages (from-to)217-291
Number of pages75
JournalFundamenta Mathematicae
Issue number3
StatePublished - 2019

Bibliographical note

Funding Information:
Proof. Appeal to Theorem 5.1 with (µ, χ, Ω) := (λ+, ℵ0, ∅) to obtain a C-sequence satisfying clause (1) of Theorem 5.4. Then, let D⃗ be the C-sequence provided by clause (2) of Theorem 5.4. Since forcing with the Acknowledgments. This work was partially supported by the Israel Science Foundation (Grant #1630/14). During the revision of this paper, the first author was supported by the Center for Absorption in Science, Ministry of Aliyah and Integration, State of Israel.

Publisher Copyright:
c Instytut Matematyczny PAN, 2019


  • And phrases: Aronszajn tree
  • C-sequence
  • Club guessing
  • Distributive tree
  • Fat set
  • Nonspecial Aronszajn tree
  • Postprocessing function
  • Square principle
  • Uniformly coherent Souslin tree
  • Walks on ordinals


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