A new model for all $C$-sequences are trivial
Assaf Rinot, Zhixing You, Jiachen Yuan
TL;DR
The paper proves that it is possible to have all $C$-sequences trivial, quantified by $χ(\kappa)=1$, while simultaneously constructing a $\kappa$-Souslin tree with full vanishing levels, i.e., $V(T)=acc(\kappa)$. The authors introduce a mutually exclusive $\mathcal{F}$-ascent path to Kunen's forcing, forming a pair $(S^\kappa_{\mathbf X}, A^\kappa_{\mathbf X})$ that is ${<}\kappa$-strategically closed and κ-cc, ensuring a well-behaved forcing extension. The main result is achieved by an Easton-support iteration over inaccessibles, enabling the coexistence of $χ(\kappa)=1$ with a club-vanishing, uniformly homogeneous $\kappa$-Souslin tree, and by lifting elementary embeddings to control the $C$-sequence structure. The findings illuminate the optimal balance between compactness and incompactness in higher cardinals and extend the landscape of models where weak compactness interacts with vanishing levels and $C$-sequences. The techniques have broader implications for the interaction of large cardinals, forcing axioms, and combinatorial set theory at uncountable cardinals.
Abstract
We construct a model in which all $C$-sequences are trivial, yet there exists a $κ$-Souslin tree with full vanishing levels. This answers a question of Lambie-Hanson and Rinot, and provides an optimal combination of compactness and incompactness. It is obtained by incorporating a so-called mutually exclusive ascent path to Kunen's original forcing construction.