On the definability of mad families of vector spaces

Abstract

We consider the definability of mad families in vector spaces of the form $\underset{n<\omega }{⨁}F$ where $F$ is a field of cardinality $\le {\aleph }_0$. We show that there is no analytic mad family of subspaces when $F={\mathbb{F}}_2$, partially answering a question of Smythe. Our proof relies on a variant of Mathias forcing restricted to a certain idempotent ultrafilter whose existence follows from Glazer's proof of Hindman's theorem.