Full 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 of Smythe (2019) about the definability of maximal almost disjoint families (mad families) of subspaces. We construct a full mad family of block subspaces in $\mathsf {ZFC}$, answering a problem by Smythe in the positive. A variant of this construction shows that there exists a completely separable mad family of block subspaces in $\mathsf{ZFC}$. We also discuss the abstract Mathias forcing introduced by Di Prisco, Mijares and Nieto (2017), and apply it to show that in Solovay’s model obtained by the collapse of a Mahlo cardinal, there are no full mad families of subspaces over $\mathbb{F}_2$.
