KöMaL problem A, 2014

Combinatorics Difficulty 6.0 National Olympiad Prove it Hungary

Let $n\ge2$ , and let $\mathcal{S}$ be a family of some subsets of $\{1,2,\ldots,n\}$ with the property that $|A\cup B\cup C\cup D|\le n-2$ for all $A,B,C,D\in\mathcal{S}$ . Show that $|\mathcal{S}|\le 2^{n-2}$ .
(CIIM6, Costa Rica)
(5 pont)

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