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A group of k people where some (n+1)-th knows all n

Source: French TST 2012

August 2, 2012
group theoryabstract algebragraph theorycombinatorics proposedcombinatorics

Problem Statement

Let nn and kk be two positive integers. Consider a group of kk people such that, for each group of nn people, there is a (n+1)(n+1)-th person that knows them all (if AA knows BB then BB knows AA). 1) If k=2n+1k=2n+1, prove that there exists a person who knows all others. 2) If k=2n+2k=2n+2, give an example of such a group in which no-one knows all others.
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