Towers and gaps at uncountable cardinals

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 $\lambda< \mathfrak p(\kappa)$. 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 $\mathfrak{p}(\kappa) = \kappa^+ < \mathfrak{p}_{cl}(\kappa) = 2^\kappa$. In contrast to earlier arguments by Shelah and Spasojevic, we achieve this by adding $\kappa$-Cohen reals and then successively diagonalising the club-filter; the latter is shown to preserve a Cohen witness to $\mathfrak{p}(\kappa) = \kappa^+$.