Largest dyadic dual VC-dimension of non-piercing families

The dyadic dual VC-dimension of a set system $\mathcal{F}$ is the largest integer $\ell$ such that there exist $\ell$ sets $F_1, F_{2}, \dots, F_\ell \in \mathcal{F}$, where every pair $\{i, j\} \in \binom{[\ell]}{2}$ 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 $k \in [\ell] \setminus \{i, j\}$. 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 P\'{a}lv\"{o}lgyi and Z\'{o}lomy.