Given a hypergraph $H=(V,E)$ and a set $X\subseteq V$ of vertices, let $int(X)$ be the number of distinct intersections of edges with $X$, i.e. $$int(X)=|\{S\subseteq X, \exists e\in E, e\cap X=S\}|.$$

$X$ is called *shattered* if $int(X)=2^{|X|}$, i.e. if $int(X)$ reaches its maximum feasible value.

Question:Is the following claim true?

Claim:If $H$ has a set $X$ with $int(X)\geq 2^{|X|}/2$, then $H$ has a shattered set of size at least $|X|/2$.

*Remarks:*

$\bullet$ The two constants "2" are somewhat arbitrary here, I really want to find out if this claim is true for some pair of (not necessarily equal) constants.

$\bullet$ The claim is true if $|X|\leq 4$ but my proof cannot be extended to higher values.

$\bullet$ My motivation comes from questions related to the *Vapnis-Chervonenkis dimension* of a hypergraph, i.e. the biggest size of a shattered set.