Six people, and a triangle you can't avoid
Asked at Jane Street, Citadel, Five Rings
At a gathering of six people, any two of them are either friends or strangers (one or the other, for every pair).
Prove that there must exist three people who are all mutual friends, or three who are all mutual strangers, no matter how the friendships are arranged.
Your answer
This one is open-ended. Work it through, then check your reasoning against the full solution.