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Folkman's number is equal to $$2 \uparrow\uparrow\uparrow (2^{901})$$ using up-arrow notation.  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 E1#1#(E901) (exact)
Fast-growing hierarchy $$f_4(f_2(892))$$