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

The answer is $\Theta (n^2)$. All four-sets containing one fixed pair form a family of sizewhose distinct intersections have size two or three. The Ray-Chaudhuri–Wilson theorem for the two allowed intersection sizes gives the matching $O(n^2)$ upper bound.