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Graham's number achieved a kind of cult status, thanks to Martin Gardner, as the largest finite number appearing in a mathematical proof. (It may no longer hold that record, but that is not my concern here.) I was surprised to learn relatively recently that it is not actually the best known bound for that particular Euclidean Ramsey problem, and that the original paper by Graham and Rothschild, which predates "Graham's number," explicitly derives a better bound. I'm left to assume that Graham later found a simpler argument that gave a weaker bound, that we now know as Graham's number.

Some time ago, before I realized the above facts, I asked Graham about his "Graham's number" proof. As I recall the conversation, he no longer had the argument at his fingertips and did not seem too interested in trying to reconstruct it. This brings me to my question:

Can someone reconstruct a simple argument for the Euclidean Ramsey problem in question that naturally yields Graham's number as an upper bound?

This would not normally be that interesting a question except that Graham's number still circulates in recreational mathematics circles, so it's a bit embarrassing if nobody knows how to "derive" it.