Definable Maximal Independent Families

Maximal independent families are combinatorial objects that have applications in various areas of mathematics. A maximal independent family can be constructed using the Axiom of Choice, and an old result of Arnold Miller shows that there are no analytic maximal independent families. We strengthen this result by showing that in the Cohen model, there are no projective maximal independent families. We also introduce a new cardinal invariant related to maximal independent families and provide some partial results about it. This is joint work with Jörg Brendle (U Kobe, Japan).