Definability of mad families of vector spaces and two local Ramsey theories

Abstract: Let $E$ be a vector space over a countable field of dimension $\aleph_0$. Two infinite-dimensional subspaces $V,W \subseteq E$ are almost disjoint if $V \cap W$ is finite-dimensional. This paper provides some improvements on results about the definability of maximal almost disjoint families (mad families) of subspaces in [17]. We show that a full mad family of block subspaces exists assuming either $\frak{p} = \max{\frak{b},\frak{s}}$ or a positive answer to a problem in [17], improving Smythe's construction assuming $\frak{p} = \frak{c}$. We also discuss the abstract Mathias forcing introduced by Di Prisco-Mijares-Nieto in [11], and apply it to show that in the Solovay's model obtained by the collapse of a Mahlo cardinal, there are no full mad families of block subspaces over $\mathbb{F}_2$.