Largest dyadic dual VC-dimension of non-piercing families

The dyadic dual VC-dimension of a set system $\mathcal{F}$ is the largest integer $\mathrm{\ell }$ such that there exist $\mathrm{\ell }$ sets $F_1,F_2,\dots ,F_{\mathrm{\ell }}\in \mathcal{F}$, where every pair $\{i,j\}\in \left(\genfrac{}{}{0px}{}{[\mathrm{\ell }]}{2}\right)$ is witnessed by an element $a_{i,j}\in F_i\cap F_j$ that does not belong to any other set $F_k$ with $k\in [\mathrm{\ell }]\setminus \{i,j\}$. In this paper, we determine the largest dyadic dual VC-dimension of a non-piercing family is exactly $4$, providing a rare example where the maximum of this parameter can be determined for a natural family arising from geometry. As an application, we give a short and direct proof that the transversal number $\tau (\mathcal{F})$ of any non-piercing family is at most $C\nu (\mathcal{F}{)}^9$, where $\nu (\mathcal{F})$ is the matching number and $C$ is a constant. This improves a recent result of Pálvölgyi and Zólomy.