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smo open 2nd round 2006 q4

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August 15, 2019
combinatoricssetset theorydouble counting

Problem Statement

Let nn be positive integer. Let S1,S2,⋯ ,SkS_1,S_2,\cdots,S_k be a collection of 2n2n-element subsets of {1,2,3,4,...,4n−1,4n}\{1,2,3,4,...,4n-1,4n\} so that Si∩SjS_{i}\cap S_{j} contains at most nn elements for all 1≤i<j≤k1\leq i<j\leq k. Show that k≤6(n+1)/2k\leq 6^{(n+1)/2}
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