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IMC 2016, Problem 4

Source: IMC 2016

July 27, 2016
IMCIMC 2016Setsset theorycollege contests

Problem Statement

Let n≥kn\ge k be positive integers, and let F\mathcal{F} be a family of finite sets with the following properties: (i) F\mathcal{F} contains at least (nk)+1\binom{n}{k}+1 distinct sets containing exactly kk elements; (ii) for any two sets A,B∈FA, B\in \mathcal{F}, their union A∪BA\cup B also belongs to F\mathcal{F}. Prove that F\mathcal{F} contains at least three sets with at least nn elements.
(Proposed by Fedor Petrov, St. Petersburg State University)
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