Towers and gaps at uncountable cardinals
Volume 257 / 2022
Abstract
Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\mathfrak p(\kappa )=\mathfrak t(\kappa )$ or there is a $(\mathfrak p(\kappa ),\lambda )$-gap of club-supported slaloms for some $\lambda \lt \mathfrak p(\kappa )$. While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah’s proof of $\mathfrak p=\mathfrak t$ to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that $\mathfrak p(\kappa )$ is always regular; the latter extends results of Garti. Finally, we turn to club variants of $\mathfrak p(\kappa )$ and present a new model for the inequality $\mathfrak {p}(\kappa ) = \kappa ^+ \lt \mathfrak {p}_{\rm cl}(\kappa ) = 2^\kappa $. In contrast to earlier arguments by Shelah and Spasojević, we achieve this by adding $\kappa $-Cohen reals and then successively diagonalizing the club filter; the latter is shown to preserve a Cohen witness to $\mathfrak {p}(\kappa ) = \kappa ^+$.
