 Large family avoiding a high intersection

If a forbidden intersection size exceeds $n/2$, every subset of size at most $\lfloor n/2\rfloor$ avoids it, including on self-intersection. This set family contains at least half the Boolean lattice. Consequently a fixed exponential bound $c^n$ with $c<2$ cannot hold uniformly as $n$ grows. Forbidden-intersection estimates below the middle rank therefore cannot simply be extrapolated to high intersections.