The extremal number of Venn diagrams

Abstract: We show that there exists an absolute constant $C>0$ such that any family $\mathcal{F}\subset {0,1}^n$ of size at least $Cn^3$ has dual VC-dimension at least 3. Equivalently, every family of size at least $Cn^3$ contains three sets such that all eight regions of their Venn diagram are non-empty. This improves upon the $Cn^{3.75}$ bound of Gupta, Lee and Li and is sharp up to the value of the constant.