PL-OM 1986 Problem 5

There is a chess tournament with $2n$ players ( $n > 1$ ). There is at most one match between each pair of players. If it is not possible to find three players who all play each other, show that there are at most $n^2$ matches. Conversely, show that if there are at most $n^2$ matches, then it is possible to arrange them so that we cannot find three players who all play each other.