Nonuniform t-intersecting family bound ($|\mathcal{A}|\le D(n,n-t)$)

If $|A\cap B|\ge t$, then $|A△B|=|A\cup B|-|A\cap B|\le n-t$. The Kleitman diametric theorem therefore gives the sharp nonuniform intersection bound. For $n-t=2q$, all sets of size at least $n-q$ attain it. For $n-t=2q+1$, take all sets of size at least $n-q$, together with all $(n-q-1)$-sets avoiding one fixed point.