John,

Will this work? The people correspond to the six positions, and the numbers say which group they are in on a given day.

This schedule is as balanced as possible: Every pair of people are together exactly four times. It consists of the twenty 3-element subsets of 1 through 6 put into the ten pairs so that subsets in the same pair have nothing in common.

In this schedule, person 1 is always in group 0. The group numbers can be changed for as many individual days as you want without affecting your requirements. The same is true of the numbering of the days.

Day 0 : (0, 0, 0, 1, 1, 1)
Day 1 : (0, 0, 1, 0, 1, 1)
Day 2 : (0, 1, 0, 1, 0, 1)
Day 3 : (0, 1, 1, 0, 0, 1)
Day 4 : (0, 0, 1, 1, 0, 1)
Day 5 : (0, 1, 0, 0, 1, 1)
Day 6 : (0, 0, 1, 1, 1, 0)
Day 7 : (0, 1, 0, 1, 1, 0)
Day 8 : (0, 1, 1, 0, 1, 0)
Day 9 : (0, 1, 1, 1, 0, 0)

--Victoria