Diana Carolina Montoya

arXiv2022

Higher Independence

We study higher analogues of the classical independence number on $ω$. For $κ$ regular uncountable, we denote by $i(κ)$ the minimal size of a maximal $κ$-independent family. We establish ZFC relations between $i(κ)$ and the standard higher analogues of some of the classical cardinal characteristics, e.g. $\mathfrak{r}(κ)\leq\mathfrak{i}(κ)$ and $\mathfrak{d}(κ)\leq\mathfrak{i}(κ)$. For $κ$ measurable, assuming that $2^κ=κ^+$ we construct a maximal $κ$-independent family which remains maximal after the $κ$-support product of $λ$ many copies of $κ$-Sacks forcing. Thus, we show the consistency of $κ^+=\mathfrak{d}(κ)=\mathfrak{i}(κ)<2^κ$. We conclude the paper with interesting open questions and discuss difficulties regarding other natural approaches to higher independence.

math.LO↗

arXiv2019

Towers and gaps at uncountable cardinals

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 $\mathfrak p(κ)=\mathfrak t(κ)$ or there is a $(\mathfrak p(κ),λ)$-gap of club-supported slaloms for some $λ< \mathfrak p(κ)$. 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 $\mathfrak p(κ)$ is always regular; the latter extends results of Garti. Finally, we turn to club variants of $\mathfrak p(κ)$ and present a new model for the inequality $\mathfrak{p}(κ) = κ^+ < \mathfrak{p}_{cl}(κ) = 2^κ$. In contrast to earlier arguments by Shelah and Spasojevic, we achieve this by adding $κ$-Cohen reals and then successively diagonalising the club-filter; the latter is shown to preserve a Cohen witness to $\mathfrak{p}(κ) = κ^+$.

math.LO↗