Higher Independence

Abstract

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