Even cross-intersection bound
= Even cross-intersection bound
{title2=$|\mathcal A||\mathcal B|\leq2^n$}
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.