I meet the following problem,
Given a set $S$ and $|S|=n, n\in \mathbf{N}^+$, $\mathcal{F_1},\mathcal{F_2}, \mathcal{F_3}, \mathcal{F_4}$ are four sub power sets defined on $S$, which only include sets sized $\frac{n}{2}$ and satisfy: $\forall i\in \{1,2,3,4\}$, $\forall A\ B\in \mathcal{F_i}, |A\cap B|\ge 2$.
$\textbf{Prove:}$ $\mathcal{F_1}\cup \mathcal{F_2}\cup \mathcal{F_3}\cup \mathcal{F_4} \neq \mathcal{F}(S)$, where $\mathcal{F}(S)$ is the power set containing all the subsets sized $\frac{n}{2}$.
According to my intuition, it may be should use Erdős-Ko-Rado theorem. Unfortunately, I can't prove that proposition and even find a counter-example.
Any help would be appreciated!