Grahams number is pretty big and cannot be written. TREE(3) is the length of the largest set of trees with 3 seeds such that the first tree can have 1 seed, second tree, 2 seeds, third tree, 3 seeds and so on such that no tree is inf-embeddable in a later tree. Is there a proof it is greater than grahams numbers?
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Yes, there is a proof, else it would referred to as a conjecture. – David Roberts Jun 04 '21 at 01:25
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What proof? I don't know any proof. – THLC Grade5 Jun 04 '21 at 10:27
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Hmm, if you mean that the proof seems to distributed between various emails and unpublished lecture notes of Harvey Friedman, and its not clear how joined up all the dots are, then I can see your point. However, I would suggest editing the question to be a reference request asking a complete proof, if this is what you want. As far as I know the proof lies in showing that $n(4)\gt \text{Graham}$ and then $TREE(3)\gt n(4)$. Some of Friedman's analysis of the fast-growing function $n$ is published (see the linked question), and he gives an estimate of $n(3)$. The rest is emails to the fom list. – David Roberts Jun 05 '21 at 00:00