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Definability of mad families of vector spaces and two local Ramsey theories

Clement Yung

Abstract

Let $E$ be a vector space over a countable field of dimension $\aleph_0$. Two infinite-dimensional subspaces $V,W \subseteq E$ are \emph{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 $\mathbb{F}_2$.

Definability of mad families of vector spaces and two local Ramsey theories

Abstract

Let be a vector space over a countable field of dimension . Two infinite-dimensional subspaces are \emph{almost disjoint} if 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 .

Paper Structure

This paper contains 23 sections, 39 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Vector spaces and block subspaces
  3. Setup and notations
  4. Intersection of block subspaces
  5. Local Ramsey theory of block subspaces
  6. Semicoideals
  7. Selectivity properties
  8. Almost disjoint families
  9. Uncountability of almost disjoint families
  10. Ideals and semicoideals generated by almost disjoint families
  11. Selectivity properties revisited
  12. Construction of mad families of subspaces
  13. Block-splitting families
  14. Proof of Theorem \ref{['thm:main.full.mad.family']}
  15. Completely separable mad family of block subspaces
  16. ...and 8 more sections

Key Result

Theorem 1.2

There exist no analytic full mad families of block subspaces.

Theorems & Definitions (118)

  • Theorem 1.2: Smythe S19
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Definition 2.1
  • Lemma 2.3
  • proof
  • Claim 1
  • proof : Proof
  • ...and 108 more