Folkman's number is equal to \(2 \uparrow\uparrow\uparrow (2^{901})\) using up-arrow notation.[1][2] In Joyce's More Generalized Exponential Notation it can be written as \(g(4,g(2,901,2),2)\).[3] It can also be written as "2 pentated to \(2^{901}\)".
History[]
Folkman's number was mentioned by Martin Gardner in the same article where he introduced the world to Graham's number, and like Graham's number it came from a problem in Ramsey theory. Jon Folkman was looking for a graph containing no \(K_4\)s that forces there to be a monochromatic \(K_3\) when it is two-colored. Folkman's number is the number of points in the graph that Folkman found.
Size[]
It can be shown that Folkman's number is between greagol and \(3 \uparrow\uparrow\uparrow\uparrow 3\). Proving the upper bound is easier: in \(3 \uparrow\uparrow\uparrow\uparrow 3 = 3 \uparrow\uparrow\uparrow (3 \uparrow\uparrow\uparrow 3)\) the base and polyponent are larger, Folkman's number \(< 3 \uparrow^{4} 3\). The lower bound is harder to prove, but it can be done fairly easily using the Knuth Arrow Theorem.
Approximations[]
Notation | Approximation |
---|---|
Chained arrow notation | \(2 \rightarrow (2 \rightarrow 901) \rightarrow 3)\) (exact) |
Hyper-E notation | E[2]1#1#(E[2]901) (exact) |
Fast-growing hierarchy | \(f_4(f_2(891))\) |
Sources[]
- ↑ Gardner, M. (1977) Mathematical games: In which joining sets of points leads into diverse (and diverting) paths Scientific American 237(5), 18-28. doi:10.1038/scientificamerican1177-18.
- ↑ [1]
- ↑ [2]