Graphs With No Pentagons Are Fundamentally Different

When you walk into a room full of people, you can speculate about all sorts of things, from political leanings to TV viewing habits. But if the room has at least six people, you can say something about them with absolute mathematical certainty, thanks to a 1930 theorem by Frank Ramsey: Among those people, there’s either a group of three who all know each other, or a group of three who have never met.

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