Nemotron-IMO-Bench 2026 C13

Prove that if $S_1,\ldots, S_p, T_1,\ldots, T_p$ are pairwise distinct subsets of $\{1,\ldots, n\}$ such that $S_i\cap T_j$ is nonempty for all $i$ and $j$, then $$p<\frac{3-\sqrt{5}}{2}\cdot 2^n.$$