Unfortunately it is known that the perfect 6 round schedule, where all players meets with 18 different people (6 as partners and 12 as opponents) does not exist. It's worth pointing out that the perfect 5 round schedule does exist - use the "20 golfers play in foursomes" schedule from
Ed Pegg's page and assign the foursomes to 2 vs 2 any way you like.
For 6 rounds, a player's partners can be all different and their opponents all different, but there may be overlap between the two groups, in other words some pairs of players meet twice, once as partners and once as opponents. The most interesting schedule I can find with this property is as follows:
( 3 7 v 14 12) ( 1 11 v 17 20) (18 16 v 9 15) (10 19 v 13 8) ( 5 2 v 6 4)
(15 20 v 12 2) ( 1 6 v 19 3) ( 4 17 v 18 13) ( 7 10 v 9 11) (14 5 v 8 16)
(11 15 v 4 8) ( 7 18 v 5 20) (12 3 v 13 19) ( 1 14 v 2 9) (17 6 v 16 10)
( 7 13 v 15 6) (19 11 v 2 16) ( 5 3 v 9 17) (20 4 v 10 14) (18 1 v 12 8)
(12 19 v 9 4) (11 6 v 18 14) ( 2 8 v 7 17) ( 5 10 v 1 15) (20 16 v 13 3)
( 9 6 v 20 8) (15 19 v 14 17) ( 5 11 v 13 12) (18 2 v 10 3) (16 1 v 7 4)
Above the players are one of two types:
Type 1 - pairs of players from the subset {3,12,13,19} are all on court together twice - once as partners and once as opponents. So a type 1 player is on court together with a total of 15 different people.
Type 2 - players from the subset {1,2,4,5,6,7,8,9,10,11,14,15,16,17,18,20} meet 18 different players. So type 2 players experience optimal mixing.
I think this it the best schedule in the sense of having the greatest possible number of Type 2 players.
