Selective Independence and $h$-Perfect Tree Forcing Notions

Abstract

Generalizing the proof for Sacks forcing, we show that the $h$-perfect tree forcing notions introduced by Goldstern, Judah and Shelah preserve selective independent families even when iterated. As a result we obtain new proofs of the consistency of $\mathfrak{i}=\mathfrak{u}<\mathrm{non}(\mathcal{N})=\mathrm{cof}(\mathcal{N})$ and $\mathfrak{i}<\mathfrak{u}=\mathrm{non}(\mathcal{N})=\mathrm{cof}(\mathcal{N})$ as well as some related results.

11 pages, intended for publication in the RIMS Set Theory 2021 Workshop Kokyuroku