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The theorem of "friends and strangers": the Ramsey number $R(3,3)=6$. Not only can the proof be understood by high-school students, proofs a proof can be discovered by students at that level via something akin to the Socratic method. First students can establish the bound $R(3,3) > 5$ by 2-coloring the edges of $K_5$:

Then they can reason through that a 2-coloring of the edges of $K_6$ must contain a monochromatic triangle, and so $R(3,3)=6$: in every group of six, three must be friends or three must be strangers.

After this exercise, an inductive proof of the 2-color version of Ramsey's theorem is in reach.

An added bonus here is that one quickly reaches the frontiers of mathematics: $R(5,5)$ is unknown! It can be a revelation to students that there is a frontier of mathematics. And then one can tell the Erdős story about $R(6,6)$, as recounted here. :-)

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The theorem of "friends and strangers": the Ramsey number $R(3,3)=6$. Not only can the proof be understood by high-school students, proofs can be discovered by students at that level via something akin to the Socratic method. First students can establish the bound $R(3,3) > 5$ by 2-coloring the edges of $K_5$:

Then they can reason through that a 2-coloring of the edges of $K_6$ must contain a monochromatic triangle, and so $R(3,3)=6$. And so R(3,3)=6$: in every group of six, three must be friends or three must be strangers. After this exercise, an inductive proof of the 2-color version of Ramsey's theorem is in reach. An added bonus here is that one quickly reaches the frontiers of mathematics:$R(5,5)$is unknown! It can be a revelation to students that there is a frontier of mathematics. And then one can tell the Erdős story about$R(6,6)\$, as recounted here. :-)

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