Improved upper and lower bounds on a geometric Ramsey problem

Abstract

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→mn, 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) © 2014.