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 $$g_n = 3\underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_{g_{n-1}}3$$, or approximately 3 → 3 → 64 → 2 in Conway's Chained Arrow Notation and {3,65,1,2} in BEAF.

In 1998, Tim Chow proved that Graham's number is much larger than the Moser.