Folkman's number is equal to \(2 \uparrow\uparrow\uparrow (2^{901})\) using up-arrow notation. [1] In Joyce's More Generalized Exponential Notation it can be written as \(g(4,g(2,901,2),2)\).[2] It can also be written as "2 pentated to \(2^{901}\)".


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.


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.


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 \(f_4(f_2(892))\)


See also

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