The vanishing levels of a tree

Joint work with Shira Yadai and Zhixing You.

Abstract. We initiate the study of the spectrum of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa$-tree $\mathbf T$. This is an invariant in the sense that if $\mathbf T$ and $\mathbf T’$ are club-isomorphic, then $V(\mathbf T)\bigtriangleup V(\mathbf T’)$ is nonstationary. Additional features of this invariant imply that the spectrum is closed under finite unions and intersections.

The set $V(\mathbf T)$ must be stationary for a homogeneous normal $\kappa$-Aronszajn tree $\mathbf T$, and if there exists a special $\kappa$-Aronszajn tree, then there exists one $\mathbf T$ that is homogeneous and satisfies that $V(\mathbf T)$ covers a club in $\kappa$. It is consistent (from large cardinals) that there is an $\aleph_2$-Souslin tree, and yet $V(\mathbf T)$ is co-stationary for every $\aleph_2$-tree $\mathbf T$. Both $V(\mathbf T)=\emptyset$ and $V(\mathbf T)=\kappa$ (modulo nonstationary) are shown to be feasible using $\kappa$-Souslin trees, even at some large cardinal close to a weakly compact. It is also possible to have a family of $2^\kappa$ many $\kappa$-Souslin trees for which the corresponding family of vanishing levels forms an antichain in the Boolean algebra of powerset of $\kappa$, modulo the nonstationary ideal.

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