Intersections of iterated shadows

We show that if $\mathcal{A}\subset \left(\genfrac{}{}{0px}{}{[n]}{n\mathrm{/}2}\right)$ with measure bounded away from zero and from one, then the $\mathrm{\Omega }(\sqrt{n})$-iterated upper shadows of $\mathcal{A}$ and ${\mathcal{A}}^c$ intersect in a set of positive measure. This confirms (in a strong form) a conjecture of Friedgut. It can be seen as a stability result for the Kruskal–Katona theorem.