A combinatorial design on a set $X$ (which I’ll call players) of size $n$ is a collection of subsets of $X$ (which I’ll call games) such that:
- Each player is in exactly $r$ games.
- Each game contains exactly $s$ players.
- Each pair of players are together in exactly $t$ games.
(actually, I think a combinatorial design is something a bit more general than this, but it’s not relevant to this question)
Obviously, an application would be scheduling a tournament in which each game involves $s$ players. There are books out there on combinatorial designs, and one of the fundamental questions is: for which values of the parameters $(n, r, s, t)$ does a combinatorial design exist?
I would like to know the answer to that question, and the related question of how we can construct such designs in practice, however, I’d like to relax the last constraint significantly to:
- Each pair of players are together in at least one game.
I can’t find anything on this version of the problem. Is it equivalent to something simpler with a different name?