Question

Suppose that $2n$ tennis players compete in a round-robin tournament. Every player has exactly one match with every other player during $2n-1$ consecutive days. Every match has a winner and a loser. Show that it is possible to select a winning player each day without selecting the same player twice. \\ \\ \textit{Hint: Remember Hall's Theorem}

Answer 1

Explanation:

given that Suppose that $2n$ tennis players compete in a round-robin tournament. Every player has exactly one match with every other player during $2n-1$ consecutive days.

this is going to be proved by contradiction

• Let there be a winning player each day where same players wins twice, let n = 3
• there are 6 tennis players and match occurs for 5days
• from hall's theorem, let set n days where less than n players wining a day
• let on player be loser which loses every single day in n days
• so, players loose to n different players in n days
• if he looses to n players then , n players are winner
• but, we stated less than n players are winners in n days which is contradiction.
• so,
• we can choose a winning players each day without selecting the same players twice.