Four-vertex traces of finite sets

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