Cofinal families of finite VC-dimension
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915530364944384 |
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| author | Ben-Neria, Omer Kaplan, Itay Peterzil, George |
| author_facet | Ben-Neria, Omer Kaplan, Itay Peterzil, George |
| contents | Given infinite cardinals $θ\leq κ$, we ask for the minimal VC-dimension of a cofinal family $\mathcal{F}\subseteq[κ]^{<θ}$. We show that for $θ=ω$ and $κ=\aleph_n$ it is consistent with ZFC that there exists such a family of VC-dimension $n+1$, which is known to be the lower bound. For $θ>ω$ we answer this question completely, demonstrating a strong dichotomy between the case of singular and regular $θ$. We furthermore answer some relative and generalized versions of the above question for singular $θ$, and answer a related question which appears in \cite{BBNKS}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cofinal families of finite VC-dimension Ben-Neria, Omer Kaplan, Itay Peterzil, George Logic Combinatorics 03E35, 03E55, 05D99 Given infinite cardinals $θ\leq κ$, we ask for the minimal VC-dimension of a cofinal family $\mathcal{F}\subseteq[κ]^{<θ}$. We show that for $θ=ω$ and $κ=\aleph_n$ it is consistent with ZFC that there exists such a family of VC-dimension $n+1$, which is known to be the lower bound. For $θ>ω$ we answer this question completely, demonstrating a strong dichotomy between the case of singular and regular $θ$. We furthermore answer some relative and generalized versions of the above question for singular $θ$, and answer a related question which appears in \cite{BBNKS}. |
| title | Cofinal families of finite VC-dimension |
| topic | Logic Combinatorics 03E35, 03E55, 05D99 |
| url | https://arxiv.org/abs/2509.24744 |