Graham's number

Graham's number is the upper bound to the solution of the now-unsolved problem, concerning multi-dimensional cubes.

"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*."

Graham's number is recursively defined as g64 in the series g0 = 4, g1 = 3 ↑↑↑↑ 3 and $$g_n = 3\underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_{g_{n-1}}3$$. In Jonathan Bowers' G functions, it is equal to G644 in base 3.

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.

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.