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 ∗ such that if all edges of the complete graph on the points { ± 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: N ∗ ≤ F ( F ( F ( F ( F ( F ( F ( 12 , 3 ) , 3 ) , 3 ) , 3 ) , 3 ) , 3 ) , 3 ) , where F ( m , n ) = 2 ↑ m n , an extremely fast-growing function. We bound 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 ) .