Improved lower bound on an Euclidean Ramsey problem

TL;DR

This paper advances the understanding of Euclidean Ramsey problems by improving the lower bound on the size of complete graphs that guarantee a monochromatic planar K_4 subgraph in two-colorings.

Contribution

The authors improve the lower bound on N^* from 11 to 13 in the Euclidean Ramsey problem for monochromatic planar K_4 subgraphs.

Findings

Lower bound N^* increased to 13

Bound remains within Graham's number N

Progress in Euclidean Ramsey theory

Abstract

It was previously shown that any two-colour colouring of K(C_n) must contain a monochromatic planar K_4 subgraph for n >= N^*, where 6 <= N^* <= N and N is Graham's number. The bound was later improved to 11 <= N^* <= N. In this article, it is improved to 13 <= N^* <= N.