$κ$-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>ω₁ the non-existence of barely independent families over ω₁.