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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]

## 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.

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2. [2]