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., r(κ) ≤ i(κ) and d(κ) ≤ i(κ). For κ measurable, assuming that 2κ = 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+ = d(κ) = i(κ) < 2κ.We conclude the paper with interesting open questions and discuss difficulties regarding other natural approaches to higher independence.