κ-Barely independent families and Tukey types of ultrafilters

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 > ω 1 implies the non-existence of barely independent families over ω 1 .