Imagine the following Problem.
The Student Union wants to organise a tournament with 2k participants ( $k \in \mathbb{N}$ ). There are to be m rounds and in each round players should be paired according to some rule s.t. no pair that already met meets again in the same tournament (i.e. it s not an elimination tournament; each player plays once in each round and gets ranked at the end or whatever ... )
The committee is concerned that after a certain number of rounds a clash will be unavoidable, so they want to come up with the clever way to pair players to avoid this.
My question is, how many legal rounds can we be sure of even if we organise it without any consideration for later rounds on an ad-hoc basis round by round. Whats the way to maximise the number of legal rounds and what is that maximum number ?
I am also very interested in the way people would model this situation and the reasoning/motivation that leads them to a proof/result.