  We show that the size of the intersection of a Hermitian variety in
$\PG(n,q^2)$, and any set satisfying an $r$-dimensional-subspace intersection
property, is congruent to 1 modulo a power of $p$. In particular, in the case
where $n=2$, if the two sets are a Hermitian unital and any other unital, the
size of the intersection is congruent to 1 modulo $\sqrt q$ or modulo
$\sqrt{pq}$. If the second unital is a Buekenhout-Metz unital, we show that the
size is congruent to 1 modulo $q$.
