Paul,

I think you are looking for a "one-factorization" of the complete graph on 28 points.
(well you don't want the whole one-factorization)

If you have 6 teams (0,1,2,3,4,5) then you can play for 5 weeks with out a repeat:

week 1 - 0:1 | 2:5 | 3:4
week 2 - 0:2 | 3:1 | 4:5
week 3 - 0:3 | 4:2 | 5:1
week 4 - 0:4 | 5:3 | 1:2
week 5 - 0:5 | 1:4 | 2:3

(then you just use however many weeks you need.)
(this is to check that I have the problem right!)

There is a rule on how to do this for any even number (it is possible to do for any even number!).
I will tell you the steps for 28 teams

1. Label your teams 0,1,...,27
   
   
2. Draw a big circle with 27 points evenly spaced on it. Label these 1 to 27. Then put a point in the center and label it 0.
   
   
3. Draw a line from 0 to 1. Draw a line from 2 to 27. Draw a line from 3 to 26. Continue drawing a line from the point labeled i to the point labeled 27-i+2 for i = 2,3,...,14 (14 will be matched with 15)
   
   
4. Each line between points represents who plays who on the first night.
   
   
5. Leave the points and the circle you drew but rotate around the circle the lines you drew. So the line from 0 to 1 becomes a line from 0 to 2. The line from 2 to 27 becomes a line from 3 to 1. The line from 3 to 26 becomes a line from 4 to 27.
   
   
6. These new lines represents who plays who on the second week.
   
   
7. Rotate the lines again to get who plays who on the third week.
   
   
8. Continue until you have all the matches you need

Example:

week 1 - 0:1 | 2:27 | 3:26 | .... |14:15
week 2 - 0:2 | 3:1 | 4:27 | .... |15:16
week 3 - 0:3 | 4:2 | 5:1 | .... |16:17
week 3 - 0:4 | 5:3 | 6:2 | .... |17:18

(Notice the pattern that 0 stays the same but each week you make the new schedule by adding 1 to each team number, being careful that you use that 27+1 = 1 not 28!)

Karen