Cofinal families of finite VC-dimension
Omer Ben-Neria, Itay Kaplan, George Peterzil
TL;DR
This work studies the minimal VC-dimension of a $\\lambda$-cofinal family $\\mathcal{F}\\subseteq [\\kappa]^{<\\theta}$, formalized as $\\mathrm{VCcof}(\\lambda,\\theta,\\kappa)$. It develops the framework of $n$-ordering systems and $\\prec\cdot$-closed sets to construct cofinal families with controlled VC-dimension, and proves a sharp dichotomy depending on whether $\\theta$ is regular or singular. The paper obtains: (i) consistency results showing $\\mathrm{VCcof}(\\omega,\\omega,\\omega_n)=n+1$ for each $n$; (ii) a dichotomy for regular $\\theta$ giving an exact threshold $\\kappa=\\theta^{+n}$ for $\\mathrm{VCcof} = n+1$; (iii) a singular-cardinal obstruction where $\\mathrm{cof}(\\theta) <\\lambda$ forces infinite VC-dimension; and (iv) a Cohen-forcing construction demonstrating the consistency of finite-closure ordering systems yielding the predicted VC-dimensions on $\\omega_n$. These results illuminate a deep regular/singular split and connect to model-theoretic concepts like IP/NIP and PAC learnability, with potential applications in constructing extensive families of low-complexity definable sets.
Abstract
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}.