Context
Graham's number is connected to the following problem in Ramsey theory:
Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then colour each of the edges of this graph either red or blue.
What is the smallest value of n for which every such colouring contains at least one single-coloured complete subgraph on 4 coplanar vertices?
In 1971, Graham and Rothschild proved that this problem has a solution, N*, and gave as a bound 6 ≤ N* ≤ N, with N being a very large, explicitly defined number:, where in Knuth's up-arrow notation. The lower bound of 6 was later improved to 11 by Geoff Exoo in 2003, and to 13 by Jerome Barkley in 2008. Thus, the best known bounds for N* are 13 ≤ N* ≤ N.
Graham's number, G, is much larger than N:, where . This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977.
Read more about this topic: Graham's Number
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