Talk held by Diana Carolina Montoya (KGRC) at the KGRC research seminar on 2018-04-19.

**Title:** On some ideals associated with independent families

**Abstract.** The concept of independence was first introduced by Fichtenholz and Kantorovic to study the space of linear functionals on the unit interval. Since then, independent families have been an important object of study in the combinatorics of the real line. Particular interest has been given, for instance, to the study of their definability properties and to their possible sizes.

In this talk we focus on two ideals which are naturally associated with independent families: The first of them is characterized by a diagonalization property, which allows us to add a maximal independent family along a finite support iteration of some ccc posets. The second ideal originates in Shelah’s proof of the consistency of $\mathfrak i\lt \mathfrak u$ (here $\mathfrak i$ and $\mathfrak u$ are the independence and ultrafilter numbers respectively). Additionally, we study the relationship between these two ideals for an arbitrary independent family $A$, and define a class of maximal independent families — which we call densely independent — for which the ideals mentioned above coincide. Building upon the techniques of Shelah we (1) characterize Sacks indestructibility for such families in terms of properties of its associated diagonalization ideal, and (2) devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family.

This is joint work with Vera Fischer.