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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2503.24207 |
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| _version_ | 1866910056679735296 |
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| author | Yung, Clement |
| author_facet | Yung, Clement |
| contents | 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 $\mathbb{F}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24207 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Full mad families of vector spaces and two local Ramsey theories Yung, Clement Logic 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 $\mathbb{F}_2$. |
| title | Full mad families of vector spaces and two local Ramsey theories |
| topic | Logic |
| url | https://arxiv.org/abs/2503.24207 |