Four-vertex traces of finite sets

Let $[n]=X_1\cup X_2\cup X_3$ be a partition with $\lfloor \frac{n}{3}\rfloor \le \mathrm{\mid }{X}_i\mathrm{\mid }\le \lceil \frac{n}{3}\rceil$ and define $\mathcal{G}=\{G\subset [n]\text{\hspace{0.17em}⁣}:\mathrm{\mid }G\cap X_i\mathrm{\mid }\le 1,1\le i\le 3\}$. It is easy to check that the trace ${\mathcal{G}}_{\mid Y}:=\{G\cap Y\text{\hspace{0.17em}⁣}:G\in \mathcal{G}\}$ satisfies $\mathrm{\mid }{\mathcal{G}}_{\mid Y}\mathrm{\mid }\le 12$ for all 4-sets $Y\subset [n]$. For $n\ge 25$ it is proven that whenever $\mathcal{F}\subset 2^{[n]}$ satisfies $\mathrm{\mid }\mathcal{F}\mathrm{\mid }>\mathrm{\mid }\mathcal{G}\mathrm{\mid }$ then $\mathrm{\mid }{\mathcal{F}}_{\mid C}\mathrm{\mid }\ge 13$ for some $C\subset [n]$, $\mathrm{\mid }C\mathrm{\mid }=4$. Several further results of a similar flavor are established as well.