Folkman's number

Folkman's number is equal to \(2 \uparrow\uparrow\uparrow (2^{901})\) using up-arrow notation.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. In Joyce's More Generalized Exponential Notation it can be written as \(g(4,g(2,901,2),2)\). 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 ==
== See also ==
*Graham's number
Category:Numbers
Category:3 entry linear
Category:Up-arrow notation level
Category:Eponymous numbers
Category:Powers of 2
Category:Pentation
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