$\kappa$-barely independent families and Tukey types of ultrafilters

Abstract

Given two infinite cardinals $\kappa$ and $\lambda$, we introduce and study the notion of a $\kappa$-barely independent family over $\lambda.$ We provide some conditions under which these types of families exist. In particular, we relate the existence of large $\kappa$-barely independent families with the generalized reaping numbers $\mathfrak{r}(\kappa,\lambda)$ and use these relations to give conditions under which every uniform ultrafilter over a given cardinal $\lambda$ is both Tukey top and has maximal character. Finally, we show that $\mathfrak{p}>\omega_1$ the non-existence of barely independent families over $\omega_1.$