= Oddtown theorem
{c}
{title2=$|\mathcal A|\leq n$}
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.
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