Problem

Source: Chinese TST 2009 1st quiz P2

Tags: ceiling function, pigeonhole principle, induction, graph theory, combinatorics proposed, combinatorics



Let $ n,k$ be given positive integers satisfying $ k\le 2n - 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m = f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.