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.