Given two infinite cardinals κ and λ, we introduce and study the notion of a κ-barely independent family over λ. We provide some conditions under which these types of families exist. In particular, we relate the existence of large κ-barely independent families with the generalized reaping numbers r (κ, λ) and use these relations to give conditions under which every uniform ultrafilter over a given cardinal λ is both Tukey top and has maximal character. Finally, we show that p>ω₁ the non-existence of barely independent families over ω₁.
Jorge Antonio Cruz Chapital (Wed,) studied this question.