Question 32
Let $k\geq1$ be an integer. A telephone provider lets each customer choose $k$ numbers for free calls: if $A$ chooses $B$'s number, calls in both directions between $A,B$ are free. Consider a group of $n$ people.
1) If $n\geq2k+2$, prove that some two people cannot communicate for free.
2) If $n=2k+1$, prove that the people can arrange their choices so every person can communicate for free with every other.
1) If $n\geq2k+2$, prove that some two people cannot communicate for free.
2) If $n=2k+1$, prove that the people can arrange their choices so every person can communicate for free with every other.