Eventown theorem ($|\mathcal{A}|\le 2^{\lfloor n/2\rfloor }$)

If every member of a set family on $[n]$ and every distinct pairwise intersection has even size, its characteristic vectors span a self-orthogonal binary subspace. The standard dot product is nondegenerate, so $W\subseteq W^{\perp }$ implies $2\dim W\le n$. Thus the set family has at most $2^{\lfloor n/2\rfloor }$ members. All unions of $\lfloor n/2\rfloor$ disjoint pairs of points attain this bound.