  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.
