Graham's Number is Less Than 2^^^6
Combinatorics
2013-08-27 v2
Abstract
In [5] Graham and Rothschild consider a geometric Ramsey problem: finding the least n such that if all edges of the complete graph on the points {+1,-1}^n are 2-colored, there exist 4 coplanar points such that the 6 edges between them are monochromatic. They give an explicit upper bound: F(F(F(F(F(F(F(12))))))), where F(m) = 2^^(m)^^3, an extremely fast-growing function. By reducing the problem to a variant of the Hales-Jewett problem, we find an upper bound which is between F(4) and F(5).
Cite
@article{arxiv.1304.6910,
title = {Graham's Number is Less Than 2^^^6},
author = {Mikhail Lavrov and Mitchell Lee and John Mackey},
journal= {arXiv preprint arXiv:1304.6910},
year = {2013}
}