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Residents of the town of Smallville are quick to judge each other: once they have met,...

Residents of the town of Smallville are quick to judge each other: once they have met, they either like each other or hate each other (and feelings are mutual). Smallville is not large; in fact, out of any three people, two have already met. Prove that in any group of 17 or more residents of the town, there are either three who all like each other or there are three who all hate each other.

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