BALKAN 1997 Problem 2

Let $S = \{A_1,A_2,\ldots ,A_k\}$ be a collection of subsets of an $n$ -element set $A$ . If for any two elements $x, y \in A$ there is a subset $A_i \in S$ containing exactly one of the two elements $x$ , $y$ , prove that $2^k\geq n$ .

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