Oddtown theorem ($|\mathcal{A}|\le n$)

If all members of a set family on $[n]$ have odd size and all distinct pairwise intersections have even size, its characteristic vectors over the finite field ${\mathbb{F}}_2$ are linearly independent. Their pairwise dot products are zero, and each vector has dot product one with itself. Taking the dot product of a linear combination with each vector makes every coefficient vanish, so there are at most $n$ vectors. The set family of all singleton sets attains this bound.