Problem

Source: French TST 2012

Tags: group theory, abstract algebra, graph theory, combinatorics proposed, combinatorics



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