Mad families of vector subspaces and the smallest nonmeager set of reals

TL;DR

This paper demonstrates that under a certain set-theoretic principle, the smallest size of a maximal almost disjoint family of vector subspaces is , and this holds in the Miller model, linking set theory and vector space combinatorics.

Contribution

It establishes a connection between a parametrized -diamond principle and the minimal size of maximal almost disjoint families of vector subspaces.

Findings

The minimal size of such families is  under the principle.

This invariant remains  in the Miller model.

The work links set-theoretic principles with properties of vector space families.

Abstract

We show that a parametrized $\diamondsuit$ principle, corresponding to the uniformity of the meager ideal, implies that the minimum cardinality of an infinite maximal almost disjoint family of block subspaces in a countable vector space is $\aleph_1$. Consequently, this cardinal invariant is $\aleph_1$ in the Miller model.