The simple version: Among 6 people, to avoid a trio of friends, no sub-group can be 3 or more. Maximize the size of the sub-groups to minimize the number of no

The simple version: Among 6 people, to avoid a trio of friends, no sub-group can be 3 or more. Maximize the size of the sub-groups to minimize the number of non-friends: 2 + 2 + 2. One from each sub-group is a trio of non friends. So either you have enough sub-groups to create a trio of non-friends, or there's a sub-group large enough to create a trio of friends.