Graham's Number is Less Than 2^^^6

TL;DR

This paper improves the upper bound for a geometric Ramsey problem involving 2-colored complete graphs on points in {-1,1}^n, showing it is less than a tower function of height 6, significantly lower than previous bounds.

Contribution

The authors reduce the problem to a variant of the Hales-Jewett problem, establishing a much tighter upper bound between F(4) and F(5).

Findings

Upper bound for the problem is less than F(4) and F(5).

Reduction to Hales-Jewett problem simplifies the analysis.

Significantly improves previous bounds for Graham's number.

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).