Improved upper and lower bounds on a geometric Ramsey problem

In Graham and Rothschild (1971), Graham and Rothschild consider a geometric Ramsey problem: finding the least N∗N∗ such that if all edges of the complete graph on the points {±1}N∗{±1}N∗ are 22-colored, there exist 44 coplanar points such that the 66 edges between them are monochromatic. They give an explicit upper bound: N∗≤F(F(F(F(F(F(F(12,3),3),3),3),3),3),3),N∗≤F(F(F(F(F(F(F(12,3),3),3),3),3),3),3), where F(m,n)=2↑mnF(m,n)=2↑mn, an extremely fast-growing function. We bound N∗N∗ between two instances of a variant of the Hales–Jewett problem, obtaining an upper bound which is less than 2↑↑↑6=F(3,6)2↑↑↑6=F(3,6).