Graham's number 
is an extremely large integer that arose as an upper
bound in a problem of Ramsey theory. Let 
be the smallest dimension 
such that every two-coloring
of the lines joining all pairs of corners of an 
-dimensional hypercube contains
a complete graph 
of one color with coplanar vertices.
Stated colloquially, consider all possible committees drawn from 
people, and assign every pair of committees one of two colors.
Each committee corresponds to a corner of the hypercube,
with coordinates indicating membership. The problem asks for the smallest 
guaranteeing four distinct committees 
, 
, 
, and 
whose six pairs have the same color and for which each person's
total number of memberships in 
and 
equals their total number in 
and 
. This condition makes the four
membership vectors the corners of a parallelogram.
The weaker condition that each person belongs to an even number of the four committees,
as stated by Hoffman (1998, p. 54), does not ensure coplanarity.
Graham and Rothschild (1971) proved that 
exists. The upper bound later popularized as Graham's number
(Gardner 1977, Lavrov et al. 2014) gives
 |
(1)
|
where Graham's number 
is recursively defined by
 |
(2)
|
and
 |
(3)
|
for 
. Here, 
is Knuth up-arrow notation.
Exoo (2003) described 
as a number often cited as the largest ever put to practical use.
In chained arrow notation, 
satisfies the inequality
 |
(4)
|
Although the full decimal expansion of 
is far too long to write down, any fixed number of its
final digits can be computed using modular arithmetic. In particular, its final ten
decimal digits are 2464195387 (OEIS A165736).
Graham and Rothschild (1971) also provided a lower bound of 6. Exoo (2003) raised the lower bound to 11 and gave experimental evidence suggesting a larger value. Barkley
(2008) raised it to 13. Lavrov et al. (2014) proved 
, and Lipka (2020) improved the upper bound
to give
 |
(5)
|
-Parameter Sets." Trans. Amer. Math. Soc. 159,
257-292, 1971.Havil, J.