Solution
= Solution
The answer is $\Theta(n^2)$. Take a <Steiner triple system>, or a partial one of quadratic size, on $\{2,\ldots,n\}$ and adjoin the fixed point $1$ to every triple. Distinct triples meet in zero or one point, so the resulting four-sets meet in one or two points. This gives $\Omega(n^2)$ members. The <Ray-Chaudhuri–Wilson theorem> with the two allowed intersection sizes gives $O(n^2)$.