Question 127
In a round robin chess tournament each player plays with every other player exactly once. The winner of each game gets $1$ point and the loser gets $0$ point. If the game ends in a tie, each player gets $0.5$ point. Given a positive integer $m$, a tournament is said to have property $P(m)$ if the following holds: among every set $S$ of $m$ players, there is one player who won all his games against the other $m-1$ players in $S$ and one player who lost all his games against the other $m-1$ players in $S$.
For a given integer $m \ge 4$, determine the minimum value of $n$ (as a function of $m$) such that the following holds: in every $n$-player round robin chess tournament with property $P(m)$, the final scores of the $n$ players are all distinct.
For a given integer $m \ge 4$, determine the minimum value of $n$ (as a function of $m$) such that the following holds: in every $n$-player round robin chess tournament with property $P(m)$, the final scores of the $n$ players are all distinct.