This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
Any group of 6 or more contains either 3 mutual friends or 3 mutual strangers.
Take a person X. Of the five other people, there must either be at least three
acquaintances of X or at least three strangers of X. Assume wlog that X has
three strangers A,B,C. Unless A,B,C is the required triad of acquaintances,
they must include a pair of strangers, wlog A,B. Then X,A,B is the required
triad of strangers, QED.