Filters, ideal independence and ideal Mr\'owka spaces

A family $\mathcal{A} \subseteq [\omega]^\omega$ such that for all finite $\{X_i\}_{i\in n}\subseteq \mathcal A$ and $A \in \mathcal{A} \setminus \{X_i\}_{i\in n}$, the set $A \setminus \bigcup_{i \in n} X_i$ is infinite, is said to be ideal independent. We prove that an ideal independent family $\mathcal{A}$ is maximal if and only if $\mathcal A$ is $\mathcal J$-completely separable and maximal $\mathcal J$-almost disjoint for a particular ideal $\mathcal J$ on $\omega$. We show that $\mathfrak{u}\leq\mathfrak{s}_{mm}$, where $\mathfrak{s}_{mm}$ is the minimal cardinality of maximal ideal independent family. This, in particular, establishes the independence of $\mathfrak{s}_{mm}$ and $\mathfrak{i}$. Given an arbitrary set $C$ of uncountable cardinals, we show how to simultaneously adjoin via forcing maximal ideal independent families of cardinality $\lambda$ for each $\lambda\in C$, thus establishing the consistency of $C\subseteq \hbox{spec}(\mathfrak{s}_{mm})$. Assuming $\mathsf{CH}$, we construct a maximal ideal independent family, which remains maximal after forcing with any proper, $^\omega\omega$-bounding, $p$-point preserving forcing notion and evaluate $\mathfrak{s}_{mm}$ in several well studied forcing extensions. We also study natural filters associated with ideal independence and introduce an analog of Mr\'owka spaces for ideal independent families.Comment: 18 Pages, subsumes arXiv:2206.1401

SELECTIVE INDEPENDENCE AND hh-PERFECT TREE FORCING NOTIONS (Recent Developments in Set Theory of the Reals)

Generalizing the proof for Sacks forcing, we show that the h-perfect tree forcing notions introduced by Goldstern, Judah and Shelah preserve selective independent families even when iterated. As a result we obtain new proofs of the consistency of i= u < non(N) = cof(N) and i < u = non(N) = cof(N) as well as some related results

SPECIALIZING WIDE ARONSZAJN TREES WITHOUT ADDING REALS (Set Theory and Infinity)

We show that under certain circumstances wide Aronszajn trees can be specialized iteratively without adding reals. We then use this fact to study forcing axioms compatible with CH and list some open problems