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.

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