TL;DR

With a few sub-optimal alternatives for tournaments with 6-players matches, we are now ready to talk about premium games.

As we already saw in Allocating games in tournaments - premium games and players, BoardGameArena has a premium program that restricts who can create a match for a limited set of games (premium games). We also saw that, for numbers solvable with affine planes, it’s easy to address the issue with $2n-1$ premium players (where $n$ is the number of participants at each match).

# What to do with tournaments having 6-players matches?

The answer depends on the model and scheduling chosen among the sub-optimal proposals in Allocating games in tournaments - 6 players matches, again.

## Ignore the (extra) round

Although we ignore the extra round, it’s interesting to take a look at it:

removed round:
()
(  1,   7,  13,  19,  25,  31,  37)
(  2,   8,  14,  20,  26,  32,  38)
(  3,   9,  15,  21,  27,  33,  39)
(  4,  10,  16,  22,  28,  34,  40)
(  5,  11,  17,  23,  29,  35,  41)
(  6,  12,  18,  24,  30,  36,  42)


This tells us a very interesting thing: taking all the players in any of the matches above, and making them premium, is sufficient to guarantee that all matches in the rounds we keep will have a premium user to create them.

How can we say this? Let’s take the first match (1, 7, …) and consider all those players premium. We know by design that any two of them will not play in the same match in any of the rounds we kept. As there are 7 of them, and each following round contains exactly 7 matches, this is actually at the same time a sufficient and necessary condition.

So there we go, we only need 7 premium users:

1 7 13 19 25 31 37


## Relax: number of participants in a match

In this case, the eighth round is the following:

round 8:
(  1,   7,  13,  19,  25,  31,  37)
(  2,   8,  14,  20,  26,  32,  38)
(  3,   9,  15,  21,  27,  33,  39)
(  4,  10,  16,  22,  28,  34,  40)
(  5,  11,  17,  23,  29,  35,  41)
(  6,  12,  18,  24,  30,  36,  42)


As we saw in the previous section, taking the whole first match players and making them premium is sufficient to address all the first seven rounds. To keep things simple, we also take the whole first column and ensure addressing this extra round too. In a twist, we note that we can take any element in any row, so we will take player 8 instead of player 2.

Hence this is a sufficient list for our purposes, with 12 elements inside:

1 3 4 5 6 7 8 13 19 25 31 37


## Relax: some pairs can face twice

In this case, the eighth round is the following:

round 8:
(       7,  13,  19,  25,  31,  37)
(       8,  14,  20,  26,  32,  38)
(  3,   9,  15,       27,  33,  39)
(  4,  10,       22,  28,  34,  40)
(  5,  11,  17,  23,       35,  41)
(  6,  12,       24,  30,  36,  42)
(  1,   2,  21,  16,  18,  29     )


Again, the whole first match plus player 1 suffices to address every other round. Again, all other matches in this round can be addressed by adding players 3 to 8, like in the previous section.

Hence this is a sufficient list for our purposes, with 12 elements inside (this is the same list as the previous section):

1 3 4 5 6 7 8 13 19 25 31 37


# That’s all folks!

What are you waiting for? Go set up a tournament!!!

If you want to take a look at all posts, here’s the list: