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arXiv · 2509.24744

Cofinal families of finite VC-dimension

Omer Ben-Neria·Itay Kaplan·George Peterzil
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Abstract

Given infinite cardinals $\theta\leq \kappa$, we ask for the minimal VC-dimension of a cofinal family $\mathcal{F}\subseteq[\kappa]^{<\theta}$. We show that for $\theta=\omega$ and $\kappa=\aleph_n$ 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 $\theta>\omega$ we answer this question completely, demonstrating a strong dichotomy between the case of singular and regular $\theta$. We furthermore answer some relative and generalized versions of the above question for singular $\theta$, and answer a related question which appears in \cite{BBNKS}.

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Omer Ben-Neria, Itay Kaplan, George Peterzil. 2025-09-29. Cofinal families of finite VC-dimension. https://arxiv.org/abs/2509.24744

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