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

TL;DR

This paper explores the construction of large classes of maximal $ heta$-independent families using large cardinal assumptions and forcing techniques, extending known results about independence families.

Contribution

It introduces methods to create proper classes of maximal $ heta$-independent families from large cardinals, expanding the scope beyond countable cases.

Findings

A single $ heta^+$-strongly compact cardinal yields a proper class of maximal $ heta$-independent families in a set-generic extension.

A proper class of measurable cardinals leads to a proper class of $ heta$ with maximal $ heta$-independent families in a class-generic extension.

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.