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Problem A. 637. (February 2015)

A. 637. Let \(\displaystyle n\) be a positive integer. Let \(\displaystyle \mathcal{F}\) be a family of sets that contains more than half of all subsets of an \(\displaystyle n\)-element set \(\displaystyle X\). Prove that from \(\displaystyle \mathcal{F}\) we can select \(\displaystyle \lceil\log_2n\rceil+1\) sets that form a separating family on \(\displaystyle X\), i.e., for any two distinct elements of \(\displaystyle X\) there is a selected set containing exactly one of the two elements.

Miklós Schweitzer competition, 2014

(5 pont)

Deadline expired on 10 March 2015.


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