Jeff,

Most pairs of players will be together 3 or 4 times over 20 weeks.

I don’t know if there is a perfect solution to your problem. There is a schedule in the archive whereby, over 6 weeks, almost every two players are together exactly once. See http://mathcentral.uregina.ca/QQ/database/QQ.09.15/h/dave1.html. In this schedule there are 4 players, each of whom missed a round with 4 (different) others, and who play their second game together on week 6. What you could do is use this schedule for the first 6 weeks, renumber the players and use it again, do that a third time, and then use the last couple of weeks to group players together who need to play together a bit more to balance things out. Renumbering can help avoid players who missed each other in the first 6 weeks missing each other again.

Good luck.
—Victoria