Full mad families of vector spaces and two local Ramsey theories

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 [18]. 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-Nieto in [12], 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 $\mathbb{F}_2$.

💡 Research Summary

The paper investigates maximal almost‑disjoint (mad) families of infinite‑dimensional subspaces of a countable‑field vector space E of dimension ℵ₀. Two subspaces are almost disjoint when their intersection is finite‑dimensional. The authors focus on block subspaces—subspaces spanned by a strictly increasing (in support) sequence of vectors—because every infinite‑dimensional subspace contains a block subspace and block subspaces enjoy strong combinatorial regularity.

First, they prove that the intersection of two block subspaces, when infinite‑dimensional, is itself a block subspace (Lemma 2.3). This corrects an earlier mistake in the literature and provides the structural foundation needed for ZFC constructions.

Using the notion of “full” introduced by Smythe—meaning that the associated semicoideal H(A) of a mad family A satisfies a pigeonhole‑type fullness property—the authors answer Smythe’s Problem 1.3 positively: they construct, in ZFC alone, a full mad family of block subspaces (Theorem 1.5). The construction relies on the inequality s ≤ non(ℳ) ≤ a_vec,F, which holds in ZFC, and on a careful transfinite recursion that adds new block vectors while preserving almost‑disjointness.

A second construction yields a completely separable mad family of block subspaces (Theorem 1.6). Complete separability means that every infinite block subspace contains a member of the family, a property previously known only under additional set‑theoretic hypotheses. The authors achieve this by strengthening the semicoideal to a selective semicoideal and exploiting the combinatorial properties of block sequences.

The latter part of the paper applies abstract Mathias forcing, as developed by Di Prisco–Mijares–Nieto, to the topological Ramsey space of block sequences over F₂. They show that in the Solovay model obtained by collapsing a Mahlo cardinal (i.e., V