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In light of the very sad news about Ron Graham's passing, I thought I'd share an interesting tidbit about his famous constant which I only learned recently (thread).
This number has captured the imaginations of many people with its unfathomable size, myself included. I distinctly remember when I was a kid first learning about it spending hours trying to wrap my mind around it and using it as inspiration for writing ever-larger numbers.
The existence of huge numbers is not in and of itself interesting, but what captivated me was how abstraction and recursion even let us _describe_ such numbers. Honestly, it was one of the first times I appreciated how powerful good definitions could be.
But this constant was not just any big number, large for the sake of being large. It had been described as the "largest number ever used in a serious mathematical proof.", with a matching Guinness Book of World Records entry and everything.
More specifically, it was used as an upper bound in a counting question about coloring edges of a high dimensional cube. The funny thing is, though, this constant wasn't actually used in the relevant paper published by Graham and Rothschild!
Instead, when Graham was explaining his work to Martin Gardner, the godfather of math popularization, he chose to use a much much larger upper bound than what he and Rothschild had used in the paper, which looked like this.
In other words, a lot of the complexity in their proof had to do with refining the upper bound to something as small as they could, but he wanted to communicate the core idea of the proof without the technical nuances of that refinement.
At first, this seems to diminish the value of Graham's constant. The number famous for being the largest number used in a serious proof was actually not used in the relevant proof!

However, I think this has a different takeaway, one which highlights how special Graham was.
This constant was not just used for a new result, it was used to make a piece of math more widely understandable. Many mathematicians would have said to Gardner "oh, the proof is a bit technical, all you need to know is that it was a big number".
But Graham was flexible enough to communicate his main idea by simplifying the argument from his paper. The goal was not just finding the tightest upper bound, it was to share an idea.

This completely changed how I see this number.
Even if there are ever bigger numbers used in proofs, TREE(n) and all, Graham's constant should hold a special place in our hearts as the number used by one genius who wanted not just to prove results, but to make sure others could see what he did.
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