Question 21
Let $n\gt 3$ be an integer and $S=\{1,\ldots,n\}$. Subsets $A_1,\ldots,A_n$ of $S$ have $|A_i|\ge2$. Suppose every two-element subset $S'\subseteq S$ is contained in exactly one $A_i$. Prove $A_i\cap A_j\ne\varnothing$ for every $i\lt j$.