  We study families $\mathcal{F}\subseteq 2^{[n]}$ with restricted
intersections and prove a conjecture of Snevily in a stronger form for large
$n$. We also obtain stability results for Kleitman's isodiametric inequality
and families with bounded set-wise differences. Our proofs introduce a new
twist to the classical linear algebra method, harnessing the non-shadows of
$\mathcal{F}$, which may be of independent interest.
