Generalized independence

TL;DR

This paper investigates various generalizations of independent families on cardinals, establishing their existence under certain set-theoretic assumptions and linking them to cardinal characteristics and ideal saturation.

Contribution

It introduces the concept of $ ext{C}$-independent families, explores their existence under $ ext{GCH}$ and $ ext{diamondsuit}^*$, and connects independence to ideal saturation.

Findings

Strongly independent families of size $2^$ exist under $ ext{diamondsuit}^*$.

Equivalence of $ ext{GCH}$ with the existence of certain strongly independent families.

Relationship between $ ext{J}$-independent families and the saturation of the ideal $ ext{J}$.

Abstract

We explore different generalizations of the classical concept of independent families on $\omega$ following the study initiated by Fisher and Montoya. We show that under $\diamondsuit^*(\kappa)$ we can get strongly independent families on $\kappa$ of size $2^\kappa$ and present an equivalence of $\mathsf{GCH}$ in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the $\mathcal{C}$-independent families, where $\mathcal{C}$ is the club filter. Also we show a relationship between the existence of $\mathcal{J}$-independent families and the saturation of the ideal $\mathcal{J}$.