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Supppose you want to organize a tournament for $n$ players split into a number of groups $g$.

You also have a fixed number of players $m$ per group.

The competition is split into a number of rounds $r$.

The situation in which two players are in the same group more than once is called an error. (If players are in the same group say 3 times you have 2 errors).

The goal is to minimise the number of errors for given $n$, $m$ and $r$, or find the maximum $r$ for given $n$ and $m$ without errors.

I currently know that there are certain $n$ for which the number of errors is 0. But I could not find a formula for these $n$ at arbitrary size.

Is there a way to find a function for these numbers or do you know in which field this topic would belong?

I would also like to find an algorithm to minimize the number of cases or be able to estimate/calculate the minimal number of errors.

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