Proper classes of maximal $\theta$-independent families from large cardinals

Abstract

While maximal independent families can be constructed from ZFC via Zorn's lemma, the presence of a maximal $\sigma$-independent family already gives an inner model with a measurable cardinal, and Kunen has shown that from a measurable cardinal one can construct a forcing extension in which there is a maximal $\sigma$-independent family. We extend this technique to construct proper classes of maximal $\theta$-independent families for various uncountable $\theta$. In the first instance, a single $\theta^+$-strongly compact cardinal has a set-generic extension with a proper class of maximal $\theta$-independent families. In the second, we take a class-generic extension of a model with a proper class of measurable cardinals to obtain a proper class of $\theta$ for which there is a maximal $\theta$-independent family.