Whenever 4*N people want to play a tournament, it is possible to set up a schedule for them. The schedule will produce a list of 4*N-1 distinct games. That's got to be the maximum possible, since that means that each player has teamed up once with every other player.
Versions of this problem have been studied for decades. A hundred years ago, the card game of Whist (similar to bridge) was so popular that many tournament schedules like this were devised and published for the convenience of clubs.
Here's the solution for 8 players:
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8 Players, 2 Courts
Court 1 2
Round ----- -----
1 AB FG CH DE
2 AC GH DB EF
3 AD HB EC FG
4 AE BC FD GH
5 AF CD GE HB
6 AG DE HF BC
7 AH EF BG CD
====================
Here's the solution for 12 players:
===========================
12 Players, 3 Courts
Court 1 2 3
Round ----- ----- -----
1 I0 45 1T 28 37 69
2 I1 56 20 39 48 7T
3 I2 67 31 4T 59 80
4 I3 78 42 50 6T 91
5 I4 89 53 61 70 T2
6 I5 9T 64 72 81 03
7 I6 T0 75 83 92 14
8 I7 01 86 94 T3 25
9 I8 12 97 T5 04 36
10 I9 23 T8 06 15 47
11 IT 34 09 17 26 58
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Here's the solution for 16 players:
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16 Players, 4 Courts
Court 1 2 3 4
Round ----- ----- ----- -----
1 AB CD EF GH IJ KL MN OP
2 EG FH AC BD MO NP IK JL
3 IL JK MP NO EH FG AD BC
4 AE IM BF JN CG KO DH LP
5 CH IN BE LO AF KP DG JM
6 BH KM CE JP DF IO AG LN
7 DE KN AH JO BG IP CF LM
8 BJ FN AI EM DL HP CK GO
9 AJ HO DK EN CL FM BI GP
10 BL EO CI HN DJ GM AK FP
11 CJ EP BK HM AL GN DI FO
12 CO GK DP HL BN FJ AM EI
13 DO FI AN GL CP EJ BM HK
14 BP GI CM FL AO HJ DN EK
15 AP FK DM GJ BO EL CN HI
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