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.