Triplet of three players is cyclic
Source: Canadian Repêchage 2012: Problem 7
May 19, 2014
combinatorics proposedcombinatorics
Problem Statement
Six tennis players gather to play in a tournament where each pair of persons play one game, with one person declared the winner and the other person the loser. A triplet of three players is said to be cyclic if wins against , wins against and wins against .
[*] (a) After the tournament, the six people are to be separated in two rooms such that none of the two rooms contains a cyclic triplet. Prove that this is always possible.[*] (b) Suppose there are instead seven people in the tournament. Is it always possible that the seven people can be separated in two rooms such that none of the two rooms contains a cyclic triplet?