Julia Millhouse | Searcharxiv

arXiv2026

Strong Projective Witnesses

We show Shelah's original creature forcing from 1984 strongly preserves tight mad families. In particular, answering questions of Fischer and Friedman and Friedman and Zdomskyy, we show the constellation $\aleph_1 = \mathfrak{a} < \mathfrak{s} = \aleph_2$ is consistent with the existence of a $\Delta_3^1$ wellorder of the reals and tight mad families of sizes $\aleph_1, \aleph_2$ which are $\Pi_1^1, \Pi_2^1$-definable, respectively. Each of these projective definitions is of minimal possible complexity.

math.LO↗

arXiv2025

Coanalytic families of functions

For Van Douwen families, maximal families of eventually different permutations and maximal ideal independent families we show that the existence of a $\Sigma^1_2$ family implies the existence of a $\Pi^1_1$ family of the same size. We also prove a similar, but slightly weaker result for generating sets of cofinitary groups.

math.LO↗