KöMaL 2008 A455

Let $H$ be a set with $n$ elements and let each of the families $\mathcal S$ and $\mathcal T$ consist of $p$ subsets of $H$ such that these 2$p$ subsets are pairwise distinct. Suppose that for every $A\in \mathcal S$ and $B\in \mathcal T$ , the sets $A$ and $B$ have at least one common element. Show that $\frac{p}{2^n}<\frac{3-\sqrt{5}}{2}$.