Friends and Enemies

Show that in each group of 6 people, there are either 3 who know each other, or 3 who do not know each other.

SOLUTION

Let’s call the people A, B, C, D, E, F. Person A either knows at least 3 among B, C, D, E, F, or does not know at least 3 among B, C, D, E, F.

Assume the first possibility – A knows B, C, D. If B and C know each other, C and D know each other, or B and D know each other, then we find a group of 3 people who know each other. Otherwise, B, C, and D form a group in which no-one knows the others.

If A doesn’t know at least 3 among B, C, D, E, F, the arguments are the same.