Four-vertex traces of finite sets

Let \([n]=X_1\cup X_2\cup X_3\) be a partition with \(\lfloor \frac{n}{3}\rfloor \le |X_i|\le \lceil \frac{n}{3}\rceil \) and define \({\mathcal {G}}=\{G\subset [n]:|G\cap X_i|\le 1, 1\le i\le 3\}\). It is easy to check that the trace \({\mathcal {G}}_{\mid Y}:=\{G\cap Y:G\in {\mathcal {G}}\}\) satisfies \(|{\mathcal {G}}_{\mid Y}|\le 12\) for all 4-sets \(Y\subset [n]\). In the present paper, we prove that if \({\mathcal {F}}\subset 2^{[n]}\) satisfies \(|{\mathcal {F}}|>|{\mathcal {G}}|\) and \(n\ge 28\), then \(|{\mathcal {F}}_{\mid C}|\ge 13\) for some \(C\subset [n]\), \(|C|=4\). Several further results of a similar flavor are established as well.