Generalized independence

Abstract

We explore different generalizations of the classical concept of independent families on ω following the study initiated by Kunen, Fischer, Eskew and Montoya. We show that under (Dℓ)κ⁎ we can get strongly κ-independent families of size 2κ and present an equivalence of 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 C-independent families, where C is the club filter. Also we show a relationship between the existence of J-independent families and the saturation of the ideal J.