Filters and Ideal Independence

A family $\mathcal{I}\subseteq [\omega {]}^{\omega }$ such that for all finite $\{X_i{\}}_{i\in n}\subseteq \mathcal{I}$ and $A\in \mathcal{I}\setminus \{X_i{\}}_{i\in n}$, the set $A\setminus {\bigcup }_{i<n}{X}_i$ is infinite, is said to be ideal independent. An ideal independent family which is maximal under inclusion is said to be a maximal ideal independent family and the least cardinality of such family is denoted ${\mathfrak{s}}_{mm}$. We show that $\mathfrak{u}\le {\mathfrak{s}}_{mm}$, which 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 $C\subseteq \text{spec}({\mathfrak{s}}_{mm})$. Assuming $\mathsf{C}\mathsf{H}$, 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.