Graham's Number

Let be the smallest dimension of a hypercube such that if the lines joining all pairs of corners are two-colored for any , a complete graph of one color with coplanar vertices will be forced. Stated colloquially, this definition is equivalent to considering every possible committee from some number of people and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find the smallest that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees (Hoffman 1998, p. 54).

An answer was proved to exist by Graham and Rothschild (1971), who also provided the best known upper bound, given by

(1)

where Graham's number is recursively defined by

(2)

and

(3)

Here, is the so-called Knuth up-arrow notation. is often cited as the largest number that has ever been put to practical use (Exoo 2003).

In chained arrow notation, satisfies the inequality

(4)

Graham and Rothschild (1971) also provided a lower limit by showing that must be at least 6. More recently, Exoo (2003) has shown that must be at least 11 and provides experimental evidence suggesting that it is actually even larger.