Even cross-intersection bound ($|\mathcal{A}||\mathcal{B}|\le 2^n$)

If every cross-intersection of two set families on $[n]$ has even size, the spans of their indicator vectors are orthogonal in ${\mathbb{F}}_2^n$. The orthogonal complement over the binary field dimension formula bounds the sum of their dimensions by $n$, giving the displayed cardinality bound.