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.

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