The chromatic number of the plane is at least 5

TL;DR

This paper introduces a new finite unit-distance graph in the plane with 1581 vertices that cannot be colored with only four colors, thus establishing a higher lower bound for the chromatic number of the plane.

Contribution

It provides the first explicit finite graph with 1581 vertices that is not 4-colorable, improving the known lower bound for the Hadwiger-Nelson problem.

Findings

The smallest known non-4-colorable unit-distance graph has 1581 vertices.

This work raises the lower bound of the plane's chromatic number from 4 to at least 5.

The constructed graph is finite and explicitly described.

Abstract

We present a family of finite unit-distance graphs in the plane that are not 4-colourable, thereby improving the lower bound of the Hadwiger-Nelson problem. The smallest such graph that we have so far discovered has 1581 vertices.