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 ${\mathbf{T}}^{\prime }$ are club-isomorphic, then $V(\mathbf{T})△V({\mathbf{T}}^{\prime })$ 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 ${\mathrm{\aleph }}_2$-Souslin tree, and yet $V(\mathbf{T})$ is co-stationary for every ${\mathrm{\aleph }}_2$-tree $\mathbf{T}$. Both $V(\mathbf{T})=\mathrm{\varnothing }$ 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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