variation on a steiner triple system

[babyteeth]

Hi there!
I have a scenario for which I cannot find a representative solution and I could use your help!
I have a group of players who want to play in a 3v3 tournament, and we want to arrange each player into rotating groups of three such that every player will play the same number of matches, play either with -or- against every other player at least once, AND minimize the total number of matches.
The current number of players is 18, but will change for each tournament, so I am looking for a generalizable solution as opposed to brute force.
Please let me know if there is any information required that I have left out. Thank you!
BT

[Ian Wakeling]

I don't think a generalizable solution exists, however what you are asking for is close to a resolvable covering.  For example see Table A.2 in this paper can potentially be converted to a minimum round schedule for 24 players on 4 courts of 3 vs 3, where all pairs of players meet at least once (either as partners or opponents).

[babyteeth]

This is very interesting, thank you! With only 18ppl at the moment, I don't know how I would compensate for so many 'ghost' players. Also, much of that math is well over my head 🤓
Perhaps I'm overthinking. I'd very much appreciate any suggestions you may have in how to approach this. I have a group of people who all play the same game, of which 10-30 of those might sign up for a tournament. It's teams of 3 vs 3, but everyone plays as individuals. Each individual player on a team gets 3 pts for the team win, 1 for team draw, 0 for team loss.
Top 6 players in points move on to a set of 3 finals matches with teams based on rank (1-6) after round one

I'm trying to find a way to assign players so that a) teams are as unique as possible. b) opponents are unique as possible. c) the system is flexible if there are odd numbers of participants. d) everyone plays the same number of matches. And e) minimizing the total number of matches, no matter how many players sign up within that range.
Also, this is a video game, so there are 'infinite' courts!

Please let me know what you think. Thank you in advance!
BT

[babyteeth]

As I was posting this, I had another thought which may be relevant. If this got bigger, I would necessarily organize this into brackets. So within bracket groupings, I would want those conditions, but no need for that across all tournament participants.

[Ian Wakeling]

I think you are asking for schedules where every pair of players meets at least once in a game (either as partners or opponents) and where the number of rounds in a minimum.  Is that right?  I am afraid I still don't have much good advice to give you here, but my earlier suggestion about the coverings was the way I would probably tackle the problem.  I would first want to find a minimal resolvable covering where all pairs meet at least once, and then I would assign the blocks of 6 players to 3 vs 3.  You would likely need software to do this but I don't have anything that will help.  I might be able to offer a solution where there are a multiple of 6 players, for example I think I can make an 18 player solution in 6 rounds, but some pairs of players will meet 3 times while other pairs only once.