The answer is $\Theta(n^2)$. All four-sets containing one fixed pair form a family of size
$$
\binom{n-2}{2}=\Theta(n^2)
$$
whose 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-109/4/i/solution