Question 61
Let $n$ and $k$ be positive integers. Consider a gathering of $k$ people such that every group of $n$ people has an $(n+1)$st person who knows all of them. Acquaintance is mutual: if $A$ knows $B$, then $B$ knows $A$.
1) If $k=2n+1$, prove that somebody knows everyone else.
2) If $k=2n+2$, give an example of such a gathering in which nobody knows everyone else.
1) If $k=2n+1$, prove that somebody knows everyone else.
2) If $k=2n+2$, give an example of such a gathering in which nobody knows everyone else.