Problem
COM-B1-M05-P016 Six People
#16
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Prove that among any \(6\) people, there are either three mutual acquaintances or three mutual strangers.
Take one person and split the others into acquaintances and strangers of that person.
Choose a person \(A\). Among the other \(5\), either \(3\) are acquainted with \(A\), or \(3\) are not. In the first case, if among those three there is an acquainted pair, together with \(A\) they form three mutual acquaintances; if not, those three are mutual strangers. The second case is analogous: if among three strangers to \(A\) there is a stranger pair, together with \(A\) they form three mutual strangers; otherwise those three are mutual acquaintances.
First Ramsey statement.