A Combinatorics Problem Involving Some Badminton Players

Question

In a badminton competition there are n players. All the players play with each other exactly once and each game has a result, i.e. win or lose. No tie is permitted. Now all the players enlist the name of some other players in the following manner - 1. He enlists the name of all the players he defeated. 2. He enlists the names of the players who are defeated by the players whom he defeated. For instance, a player A enlists the names of B and C if A defeats B and B defeats C. Prove that there exists at least one such player A who has enlisted the names of all the other players.

Answer 1

If the problem is:

Prove that there exists exactly one such player A.

Then it contradicts the question statement:

... there are n players ... all the players enlist the name of some other players ...

If we add in a condition like a unique maximum number of enlisted players, this seems to be a counterexample:

Here:

• A enlists B and C,
• B enlists C and A, and
• C enlists A and B.