Cofinal families of finite VC-dimension

Given infinite cardinals $\theta \le \kappa$, we ask for the minimal VC-dimension of a cofinal family $\mathcal{F}\subseteq [\kappa {]}^{<\theta }$. We show that for $\theta =\omega$ and $\kappa ={\mathrm{\aleph }}_n$ 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}.