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

TL;DR

This paper introduces the concept of $$-barely independent families over a cardinal and explores their existence, relating them to ultrafilter properties and generalized reaping numbers, with implications for Tukey types.

Contribution

It defines $$-barely independent families and establishes their connection to ultrafilter Tukey types and reaping numbers, providing new insights into ultrafilter structure.

Findings

Existence conditions for $$-barely independent families.

Relations between these families and generalized reaping numbers.

Non-existence of such families over $$ when $>_1$.

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 $\omega_1.$