Generalized independence

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 ${♢}^{\ast }(\kappa )$ 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 $\mathcal{J}$.