Graham's number

Graham's number is a proven upper bound to the solution of the following unsolved problem in Ramsey Theory:

"Let N* be the smallest dimension n of a hypercube such that if the lines joining all pairs of corners are two-colored for any n ≥ N*, a complete graph K4 of one color with coplanar vertices will be forced. Find N*."

The upper bound of the problem has since been reduced to \(F^{7}(12)\), where \(F(n) = 2 \uparrow^{n} 3\).

Graham's number is celebrated as the largest number ever used in a mathematical proof. Although it is much smaller than many other googologisms (such as tetratri), it is likely the largest one to have any sort of mathematical use.

Definition
Define the series g0 = 4, g1 = 3 ↑↑↑↑ 3 (using Arrow Notation), and \(g_n = 3\underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_{g_{n-1}}3\). Graham's number is g64.

Since g0 is 4 and not 3, Graham's number cannot be expressed efficiently with most major googological functions. However, it is reasonably close to 3 → 3 → 64 → 2 in Conway's Chained Arrow Notation and {3,65,1,2} in BEAF. In Jonathan Bowers' G functions, it is exactly equal to G644 or GGG...GGG4 (64 G's) in base 3.

Tim Chow proved that Graham's number is much larger than the Moser. The proof hinges on the fact that, using Steinhaus-Moser Notation, n in a (k + 2)-gon is less than \(n\underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_{2k-1}n\). He sent the proof to Susan Stepney on July 7, 1998. Coincidentally, Stepney was sent a similar proof by Todd Cesere several days later.