Full mad families of vector spaces and two local Ramsey theories

Abstract

Let E be a vector space over a countable field of dimension ℵ0. Two infinite-dimensional subspaces V,W⊆E are almost disjoint if V∩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 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 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 F2.