  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.
