Independent families in the countable and the uncountable

Independent families on $\omega$ are families of infinite sets of integers with the property that for any two finite subfamilies $A$ and $B$ the set $\bigcap A\backslash \bigcup B$ is infinite. Of particular interest are the sets of the possible cardinalities of maximal independent families, which we refer to as the spectrum of independence. Even though we do have the tools to control the spectrum of independence at $\omega$ (at least to a  large extent), there are many relevant questions regarding higher counterparts of independence in generalised Baire spaces, which remain widely open.