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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 points)

Deadline expired on 10 March 2015.

Statistics on problem A. 637.
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• Problems in Mathematics of KöMaL, February 2015

•  Támogatóink: Morgan Stanley