A small 6-chromatic two-distance graph in the plane

TL;DR

This paper presents a simple proof establishing a lower bound on the chromatic number of the Euclidean plane when two distances are forbidden, using a small graph with only 16 vertices.

Contribution

It introduces a new, straightforward proof technique for the two-distance chromatic number problem with a minimal 16-vertex graph.

Findings

Established a lower bound for the plane's chromatic number with two forbidden distances

Constructed a 16-vertex graph demonstrating the bound

Simplified the proof method for this geometric coloring problem

Abstract

We give a new, simple proof for the lower bound of the chromatic number of the Euclidean plane with two forbidden distances, based on a graph with only 16 vertices.