Half-power-set trace growth forces finite shattering ($|{\mathcal{F}}_{|Y}|\ge 2^{2k-1},|Y|=2k⟹\exists Z\subseteq Y,|Z|=k$)

If no $k$-set were shattered, the Sauer-Shelah lemma would bound the number of traces by ${\sum }_{i<k}\left(\genfrac{}{}{0ex}{}{2k}{i}\right)=(2^{2k}-(\genfrac{}{}{0ex}{}{2k}{k}))/2<2^{2k-1}$. This strict middle-layer gap contradicts the displayed lower bound. Therefore a family with such traces at every even size has unbounded finite VC dimension.