**November 13**

**Diana Montoya**,
University of Vienna

**Independence and uncountable cardinals**

The classical concept of independence, first introduced by Fichtenholz and Kantorovic has been of interest within the study of combinatorics of the subsets of the real line. In particular the study of the cardinal characteristic $\mathfrak{i}$ defined as the minimum size of a maximal independent family of subsets of $\omega.$ In the first part of the talk, we will review the basic theory, as well as the most important results regarding the independence number. We will also point out our construction of a poset $\mathbb{P}$ forcing a maximal independent family of minimal size which turns out to be indestructible after forcing with a countable support iteration of Sacks forcing.

In the second part, we will talk about the generalization (or possible generalizations) of the concept of independence in the generalized Baire spaces, i.e. within the space $\kappa^\kappa$ when $\kappa$ is a regular uncountable cardinal and the new challenges this generalization entails. Moreover, for a specific version of generalized independence, we can have an analogous result to the one mentioned in the paragraph above.

This is joint work with Vera Fischer.