# Further improving of upper bound on a geometric Ramsey problem

## TL;DR

This paper refines the upper bound on the dimension needed to guarantee a monochromatic coplanar 4-vertex clique in any 2-coloring of the hypercube's edges, improving previous bounds significantly.

## Contribution

It improves the known upper bound on the geometric Ramsey problem from a triple exponential tower to a lower triple exponential tower.

## Key findings

- Reduced the upper bound from 2↑↑↑6 to 2↑↑↑5.
- Provided a tighter estimate for the minimal dimension n.
- Revisited and improved upon previous combinatorial bounds.

## 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 $n< 2\uparrow\uparrow\uparrow 5$