Steve,

There are 153 different pairs of golfers. Over the 5 rounds it would be possible to
cover at most 120 of them. That means that it isn't possible for every two people
to have a rounds together.

Here is one way that is as good as you can do. Tthe setup might seem strange. Start
with the numbers 1, 2, ..., 25 in a 5x5 grid:

1	2	3	4	5
6	7	8	9	10
11	12	13	14	15
16	17	18	19	20
21	22	23	24	25

Then partition them into six groups of five -- the rows and:

1. the columns
   
   
2. 1, 7, 13, 19, 25; 2, 8, 14, 20, 21; 3, 9, 15, 16, 22; 4, 10, 11, 17, 23; 5, 6, 12, 18, 24
   
   
3. 1, 8, 15, 17, 24; 2, 9, 11, 18, 25; 3, 10, 13, 20, 22; 4, 6, 13, 20, 23; 5, 7, 14, 17, 24
   
   
4. 1, 9, 12, 20, 23; 2, 10, 13, 16, 24; 3, 6, 14, 17, 25; 4, 7, 15, 18, 21; 5, 8, 11, 19, 22
   
   
5. 1, 10, 14, 18, 22; 2, 6, 15, 19, 24; 3, 7, 11, 20, 24; 4, 8, 12, 16, 25; 5, 9, 13, 17, 21

Forget about the rows and focus on (1), (2), ..., (5). These will eventually be the days. First delete the numbers 21, 22, 23, 24, 25 wherever they occur. This is a schedule for 20 golfers in 5 foursomes over 5 days with no repeated pairs. (The pairs that don't play together are the ones where the golfers are in the same row if the grid.)

Finally, delete the numbers 19 and 20. This is the schedule you want. Again, there are no repeated pairs.

Enjoy!

Victoria