# Disproving or proving claim that if VCdim is "n" then it is possible that a set of smaller size is not shattered

Today in the lecture the lecturer said something I found peculiar, and I felt very inconvenient when I heard it: He claimed, that if the maximal VCdim of some hypothesis class is $$n\in\mathbb N$$, then it is possible that there is some $$i such that for every subset C of size i the subset C is not shattered. Is his claim true? I thought that we can take some subset of size $$i,\forall i\in [n]$$of the set C* which satisfies the condition for the case where $$|C|=n$$, and they will shatter as well. Am I missing something?

I agree, the claim as written is incorrect. If $$C^*$$ is shattered by $$\mathcal{H}$$, and $$C\subseteq C^*$$, then $$C$$ is also shattered by $$\mathcal{H}$$; to be possibly over-pedantic:
For each $$B\subseteq C$$, because then also $$B\subseteq C^*$$, we have that there is an $$H\in\mathcal{H}$$ such that $$C^*\cap H=B$$. Note that this implies $$B\subseteq H$$. Now, $$C\subseteq C^*$$ implies that $$C\cap H\subseteq C^*\cap H$$, so we have $$C\cap H\subseteq B$$. And having both $$B\subseteq C$$ and $$B\subseteq H$$ implies that $$B\subseteq C\cap H$$. So $$B=C\cap H$$.