A solution to Roitman's problem

TL;DR

This paper constructs a forcing that destroys a given maximal almost disjoint family while preserving certain ultrafilters and combinatorial properties, solving Roitman's problem for the case of  and related questions.

Contribution

It introduces a new construction technique for partial orders using ladder systems and trees of normed creatures, enabling solutions to longstanding set-theoretic problems.

Findings

Constructed a forcing with Axiom A and -properness that destroys a mad family.

Solved Roitman's problem for  using countable support iteration of the new forcing.

Established the relative consistency of  < a with a dominating set of size  and a minimal mad family of size .

Abstract

We answer Question~3.2 from Shelah \cite{Sh:666}: Given a maximal almost disjoint (mad) family $\mathcal A$ of size $\aleph_1$, we construct a forcing ${\mathbb Q}(\mathcal A)$ that has Axiom A, is ${}^\omega \omega$-bounding, preserves selective ultrafilters, has the $\aleph_2$-properness isomorphism condition (p.i.c.), and destroys the mad family $\mathcal A$. We develop a new construction technique for partial orders, combining ladder systems for $\omega_1$ with trees of normed creatures. Countable support iteration of the new kind of iterands solves Roitman's problem in the case of $d=\aleph_1$ and also simultaneously the open question about the relative consistency of $u = \aleph_1 < a$: It is consistent relative to ZFC that there is a dominating set of size $\aleph_1$ and a selective ultrafilter with character $\aleph_1$ and the minimal size of a mad family is $\aleph_2$, like the continuum.