Jim,
This is an extremely well researched area of mathematics. If there are n particpating players, playing in g groups of p players each over r rounds then there are many social rectangles that are possible. I have listed them below classified by p. For a level of p, the first member in the series of schedules is always a social square (g=p). Schedules for p=2 are exactly the same as the all-plays-all round robin. Note for p=2 and p=3 (threesome tournaments) all members of the series of schedules are known to exist, so I have only listed the first 6 possiblilities below. For p=4 and p=5, I have listed the known schedules up to 100 participants, so note in particular that there is no entry for n=45, p=5, this schedule may indeed be possible, but nobody has managed to prove it one way or the other.
For both examples you give in your message, schedules do exist. If you search the message board for foursomes schedules you will find several which have the never-play-anyone-on-their-own-team property.
Ian.
n g p r
4 2 2 3
6 3 2 5
8 4 2 7
10 5 2 9
12 6 2 11
14 7 2 13
9 3 3 4
15 5 3 7
21 7 3 10
27 9 3 13
33 11 3 16
39 13 3 19
16 4 4 5
28 7 4 9
40 10 4 13
52 13 4 17
64 16 4 21
76 19 4 25
88 22 4 29
100 25 4 33
25 5 5 6
65 13 5 16
85 17 5 21
