Answer to old chestnut: friends
Choose one person, call him A.
Each other person is either a friend or stranger to A, so among the other five there must be at least three friends or three strangers.
If there are three strangers to A, let them be B, C, and D. Either B, C, and D are mutual friends, or one pair of them are strangers; this pair plus A make three mutual strangers.
If there are three friends of A, the argument is similar. Let the friends be B, C, and D. Either these three are mutual strangers, or one pair of them are friends. Then that pair plus A are three mutual friends.