Cofinal families of finite VC-dimension

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}.