You can use up all 150 slots, with exactly 8 people per team, as follows: Day 1: Team 1 {1,2,3,4,5,6,7,9} Team 2 {1,2,3,4,5,6,7,9} Team 3 {1,2,3,4,5,6,7,8} Team 4 {1,2,4,6,7,10} Team 5 {1,2,3,4,5,6,7,8} Team 6 {1,2,3,4,5,6,7,9} Team 7 {1,2,3,4,5,6,7,10} Team 8 {1,2,3,4,6,9,10} Team 9 {1,2,3,4,5,6,7,10} Team 10 {1,2,3,4,5,6,7,10} Team 11 {1,2,3,4,5,6,7} Team 12 {1,2,3,4,5,6,7,8} Team 13 {2,3,4,5,6,7,9} Team 14 {1,2,3,4,6,7,9} Team 15 {1,2,3,4,5,7} Team 16 {1,2,4,5,6,7,8} Team 17 {1,2,3,5,6,7,9} Team 18 {1,2,3,4,5,6,7,10} Team 19 {1,2,3,4,5,6,7,8} Team 20 {1,2,3,4,5,6,7,10} Day 2: Team 1 {2,3,4,5,6,7,9} Team 2 {1,2,3,5,7,9} Team 3 {1,2,3,4,5,6,7,8} Team 4 {1,2,3,4,5,6,7,10} Team 5 {1,2,3,4,5,7,8} Team 6 {1,2,3,4,6,7,9} Team 7 {1,2,3,5,7,10} Team 8 {1,2,3,4,5,6,9,10} Team 9 {1,2,3,4,5,6,7,10} Team 10 {1,2,3,4,5,6,7,10} Team 11 {1,2,3,4,5,7,9} Team 12 {1,2,3,4,5,6,7,8} Team 13 {1,2,3,4,5,7,9} Team 14 {1,2,3,4,5,6,7,9} Team 15 {1,2,3,4,5,7,8} Team 16 {1,2,3,4,5,6,7,8} Team 17 {1,2,3,4,5,6,7,9} Team 18 {1,2,3,4,5,6,7,10} Team 19 {1,2,3,4,5,6,7,8} Team 20 {1,2,3,4,5,6,7,10} Day 3: Team 1 {1,2,3,4,5,7,9} Team 2 {1,2,3,4,5,6,7,9} Team 3 {1,2,3,4,5,6,7,8} Team 4 {1,2,3,4,5,6,7,10} Team 5 {1,2,3,4,5,6,7,8} Team 6 {1,2,3,5,6,7,9} Team 7 {1,3,4,5,6,7,10} Team 8 {2,3,4,5,6,9,10} Team 9 {1,2,3,4,5,7,10} Team 10 {1,2,3,4,5,6,7,10} Team 11 {1,2,3,4,5,7,9} Team 12 {1,2,3,4,5,6,7,8} Team 13 {1,2,3,4,5,6,7,9} Team 14 {1,2,3,4,5,6,7,9} Team 15 {1,3,4,5,6,7,8} Team 16 {1,3,4,5,6,7,8} Team 17 {2,3,4,5,6,7,9} Team 18 {1,2,3,4,5,6,7,10} Team 19 {2,3,4,5,6,7,8} Team 20 {1,2,3,4,5,6,7,10} Day 4: Team 1 {1,2,3,4,5,6,7,9} Team 2 {1,2,3,4,5,6,7,9} Team 3 {1,2,3,4,5,6,7,8} Team 4 {1,2,3,4,5,6,7,10} Team 5 {1,2,3,4,5,6,7,8} Team 6 {1,2,3,4,5,6,7,9} Team 7 {1,3,4,5,6,7,10} Team 8 {1,2,5,6,9,10} Team 9 {3,4,5,6,7} Team 10 {1,2,3,4,5,6,7,10} Team 11 {1,2,4,5,6,7,9} Team 12 {1,2,3,4,5,6,7,8} Team 13 {1,2,3,4,5,7,9} Team 14 {1,2,4,5,6,7,9} Team 15 {1,3,4,5,6,7,8} Team 16 {1,2,3,4,5,6,7,8} Team 17 {1,2,3,4,5,6,7,9} Team 18 {1,2,3,4,5,6,7,10} Team 19 {1,2,3,4,5,6,7,8} Team 20 {1,2,3,4,5,6,7,10} Day 5: Team 1 {1,3,4,5,7,9} Team 2 {1,2,3,4,5,6,7,9} Team 3 {1,2,3,4,5,6,7,8} Team 4 {1,2,3,4,5,6,7,10} Team 5 {1,2,4,5,6,7,8} Team 6 {2,3,4,5,6,7,9} Team 7 {1,2,3,4,5,6,7,10} Team 8 {1,2,3,4,5,6,9} Team 9 {1,2,3,4,5,6,7,10} Team 10 {2,3,4,5,6,7,10} Team 11 {1,2,3,4,5,6,7,9} Team 12 {1,2,4,5,6,7,8} Team 13 {1,2,3,4,5,6,7,9} Team 14 {1,2,3,4,5,7,9} Team 15 {1,2,4,5,6,7,8} Team 16 {1,2,3,4,5,6,7,8} Team 17 {1,2,3,4,5,6,7,9} Team 18 {1,2,3,4,5,6,7,10} Team 19 {1,2,3,4,5,6,7} Team 20 {1,2,3,4,5,6,7,10} I obtained the solution via integer linear programming, and 8 is the minimum possible value of the largest team.