Graham's number record

Graham on Graham's number

F(2,n) = F(1,F(2,n-1)) = 2^F(2,n-1)
== 2^..4 :n-2 = 2^^n
F(3,n) = F(2,F(3,n-1)) = 2^^F(3,n-1)
== 2^^..4 :n-2 = 2^^^n

F(m,n) = 2^^..n ^:m = 2→n→m

a→b→c+1→2 = a→b→(a→b→c→2)
== a→b→(..a→b→1→2.) :c:
= a→b→(..a^b.) :c:

N* ≤ 2→3→(..12.) :7:
< 4→2→8→2 == 4→2→(..16.) :7:

Graham's number record explained by Gardner
GN' ≤ 3→3→(..4.) :64:
< 2→3→65→2 = 2→3→(..8.) :64:

Graham & Rothschild's original estimate was less
GN* ≤ 2→3→(..12.) :7:
< 4→2→8→2 < 2→3→9→2

New bounds for Graham's number of hypercube dimensions
13 ≤ GN ≤ 2→6→3 [as of September 2014]

This article uses → Conway's chained arrow notation
Martin Gardner "Mathematical Games"
Scientific American, Nov. 1977, pp.18-28
R.L. Graham & B.L. Rothschild "Ramsey's theorem for n-parameter sets"
Transactions of the American Mathematical Society, Vol.159, Sept. 1971, pp.257-292
Norris McWhirter "Guinness book of World Records"
Giant 1980 super-edition, p.193, Numeration > Numbers > Highest
Jerome Barkley "Improved lower bound on an Euclidean Ramsey problem"
Lavrov, Lee, Mackey "Graham's Number is Less Than 2^^^6"
Konstantinos Tyros "On colorings of variable words"
VIDEO Ron Graham explains bounds for Graham's Number
Photograph: Ron Graham juggles 4 balls, by Peter Vidor

Presented as research material on the topic of Big numbers for which Graham's number(s) is an ongoing inspiration.