Selective Independence and $h$-Perfect Tree Forcing Notions

TL;DR

This paper extends the preservation of selective independent families under $h$-perfect tree forcing notions, providing new consistency results for certain cardinal characteristics related to independence, ultrafilters, and null sets.

Contribution

It generalizes the preservation proof for Sacks forcing to $h$-perfect tree forcing notions, leading to new consistency results for cardinal invariants.

Findings

Preserves selective independent families under $h$-perfect tree forcing

Provides new consistency proofs for $rak{i}=rak{u}< on( )$ and $rak{i}<rak{u}= on( )$

Extends known results from Sacks forcing to a broader class of 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.