Show that in each group of 6 people, there are either 3 which know each other, or 3 which do not know each other.
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 doesnt 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 which know each other. Otherwise, B, C and D form a group in which noone knows the others.
If A doesn't know at least 3 among B, C, D, E, F, the arguments are the same.