Further improving of upper bound on a geometric Ramsey problem

Abstract

We consider following geometric Ramsey problem: find the least dimension $n$ such that for any 2-coloring of edges of complete graph on the points $\{\pm 1\}^n$ there exists 4-vertex coplanar monochromatic clique. Problem was first analyzed by Graham and Rothschild and they gave an upper bound: $n\le F(F(F(F(F(F(F(12)))))))$, where $F(m) = 2\uparrow^m3$. In 2014 Lavrov, Lee and Mackey greatly improved this result by giving upper bound $n< 2\uparrow\uparrow\uparrow 6 < F(5)$. In this paper we revisit their estimates and reduce upper bound to $n< 2\uparrow\uparrow\uparrow 5$