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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.07836 |
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| _version_ | 1866909541153636352 |
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| author | Yung, Clement |
| author_facet | Yung, Clement |
| contents | Call a subset of $\mathbf{FIN}_k$ small if it does not contain a copy of $\langle{A\rangle}$ for some infinite block sequence $A \in \mathbf{FIN}_k^{[\infty]}$. Gowers' $\mathbf{FIN}_k$ theorem asserts that the set of small subsets of $\mathbf{FIN}_k$ forms an ideal, so it is sensible to consider almost disjoint families of $\mathbf{FIN}_k$ with respect to the ideal of small subsets of $\mathbf{FIN}_k$. We shall show that $\mathfrak{a}_{\mathbf{FIN}_k}$, the smallest possible cardinality of an infinite mad family of $\mathbf{FIN}_k$, is uncountable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mad families of Gowers' infinite block sequences Yung, Clement Logic Combinatorics Call a subset of $\mathbf{FIN}_k$ small if it does not contain a copy of $\langle{A\rangle}$ for some infinite block sequence $A \in \mathbf{FIN}_k^{[\infty]}$. Gowers' $\mathbf{FIN}_k$ theorem asserts that the set of small subsets of $\mathbf{FIN}_k$ forms an ideal, so it is sensible to consider almost disjoint families of $\mathbf{FIN}_k$ with respect to the ideal of small subsets of $\mathbf{FIN}_k$. We shall show that $\mathfrak{a}_{\mathbf{FIN}_k}$, the smallest possible cardinality of an infinite mad family of $\mathbf{FIN}_k$, is uncountable. |
| title | Mad families of Gowers' infinite block sequences |
| topic | Logic Combinatorics |
| url | https://arxiv.org/abs/2402.07836 |