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Say for a permutation group $G$ over $n$ that a set $S\subset \{1,\ldots,n\}$ is G-stabbed by $X\subset \{1,\ldots,n\}$ if for every $g\in G$ we have $gS\cap X\ne \emptyset$. Is there for every $|S|...

Say that $S\subset \mathbb Z_n$ is stabbed by $X\subset \mathbb Z_n$ if for every $t$ we have $(S+t)\cap X\ne \emptyset$. Is there for every $|S|=4$ an $|X|<n/2$ that stabs it? My motivation ...