I added a nice graph of K(6) I just created in Mathematica to help clarify. You may suppose the green triangle to be three mutual strangers, while the red triangle represents three mutual friends.

Also, I might add that** this problem may be viewed as “one of avoidance” **— that is, in a certain sense, one cannot “avoid” having either three friends or three strangers, if there are six people, as one of these two must be true. This applies to all the Ramsey type problems — they may **all be viewed as problems of inevitability and hence a deep unexpected structure within finite sets** — that if we have a finite graph of a given size, then** we must have a given or unavoidable colorability “structure” with that graph — it simply cannot be avoided.**