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Can someone give me an idea of how R.Graham reached Graham's Number as an upper bound on the solution of the related problem ? Thanks !

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Because it was what he needed for his proof to work? Is there something more than that which you wanted? –  Thomas Andrews Jun 26 '12 at 18:25
I would think the way to answer this question would be to read Graham's paper. Have you tried this? –  Gerry Myerson Jun 27 '12 at 1:09
I guess nobody knows: mathoverflow.net/questions/117006/… –  Andres Caicedo Jan 3 '13 at 6:54
John Baez has posted an update on the link above. –  Andres Caicedo Jan 11 '13 at 17:37
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