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This sounds like the setup to Graham’s Number. Are they connected, and if so, what’s special about Graham’s Number as compared to other Ramsey numbers?

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Graham's number does indeed come out of Ramsey theory. https://en.wikipedia.org/wiki/Graham%27s_number#Context should tell you most of what you're looking to know.

Famed starcraft and hearthstone player Day9 has a great video about Graham's number https://www.youtube.com/watch?v=1N6cOC2P8fQ

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