It's known that only 5 rounds are possible (http://www.mathpuzzle.com/MAA/54-Golf%20Tournaments/mathgames_08_14_07.html) without repeating pairs of players within foursomes.
There is a solution below for six rounds, where six pairs of players repeat.
( 6 14 13 20) ( 4 5 11 16) (19 2 12 9) (18 15 1 7) (10 17 3 8)
(12 7 10 11) (17 1 14 5) (15 9 6 3) (18 4 2 20) (13 19 8 16)
( 7 10 19 14) ( 2 8 5 15) (20 17 4 9) ( 3 18 11 13) ( 1 16 6 12)
(13 2 10 1) (20 12 5 3) ( 8 4 7 6) (14 11 15 19) ( 9 16 18 17)
( 6 5 18 19) ( 4 10 9 14) ( 8 1 20 11) (12 13 17 15) ( 7 3 16 2)
(14 12 8 18) ( 5 13 9 7) (16 20 15 10) ( 1 3 19 4) (11 6 17 2)