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Subsets of \{1, 2,\ldots, n\}

Source: Romanian TST 4 2008, Problem 3

June 13, 2008
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Problem Statement

Let n≥3 n \geq 3 be a positive integer and let m \geq 2^{n\minus{}1}\plus{}1. Prove that for each family of nonzero distinct subsets (Aj)j∈1,m‾ (A_j)_{j \in \overline{1, m}} of {1,2,...,n} \{1, 2, ..., n\} there exist i i, j j, k k such that A_i \cup A_j \equal{} A_k.
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