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Distributive Aronszajn trees. (English) Zbl 1472.03043

Summary: S. Ben-David and S. Shelah [Isr. J. Math. 53, 93–96 (1986; Zbl 0617.03026)] 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 \(\square (\lambda ^+,{<}\lambda)\). As \(\square (\lambda ^+,{<}\lambda)\) 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 [loc. cit.], 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.

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
03E65 Other set-theoretic hypotheses and axioms

Citations:

Zbl 0617.03026
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References:

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