Suppose that we have a family $\mathcal{F}$ of subsets of $[4n]$ such that $|A|=2n$ for all $A\in \mathcal{F}$ and for every $A,B\in \mathcal{F}$ we have $|A\cap B|\ge 2$. Then$$|\mathcal{F}|\le \frac{1}{2}\left(\left(\genfrac{}{}{0ex}{}{4n}{2n}\right)-{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}^2\right).$$

unreviewedOpen. Worked here; no verified result yet.

combinatorics · solved · prize $500 · 0 attempts