Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 109 / 4 / ii / Solution

The answer is $\Theta (n)$. The familyhas size $n-3$ and pairwise intersection three. For the upper bound, work modulo two. Every member has size $0$ modulo two, while every allowed intersection has residue $1$. The Frankl-Wilson theorem with $L=\{1\}$ gives $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{n}{1}\right)=n$.