PR 4 # 26
 

In any gathering of six people prove that either three are mutually acquainted or three are mutually unacquainted.

[spoiler]Label the persons A to F. For A there are either at least three persons he knows or three he does not know. Let us assume that he knows at least three. If either of these three persons (labeled conveniently B, C, D) knows another of these three, we will have three persons that mutually know each other. So B, C and D do not know each other. But know we have three persons that mutually do not know each other.
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