Madness in vector spaces

TL;DR

This paper explores the size and definability of maximal almost disjoint families of subspaces in countable vector spaces, revealing independence results and applying local Ramsey theory to understand their complexity.

Contribution

It demonstrates that the minimal size of such families is independent of ZFC and shows the spectrum of their cardinalities can be arbitrarily large, extending known results.

Findings

Minimal size of mad families is independent of ZFC.

Spectrum of cardinalities can be arbitrarily large.

Partial results on definability using local Ramsey theory.

Abstract

We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the "spectrum" of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on $\omega$ . We apply the author's local Ramsey theory for vector spaces to give partial results concerning their definability.