A conditional approach for monochromatic unit distance in a plane for four and five coloring

TL;DR

This paper introduces a measure theoretic approach to demonstrate the existence of finite unit-distance graphs in the plane that cannot be colored with four or five colors, advancing understanding of graph colorability in geometric settings.

Contribution

It presents a novel measure theoretic method to establish the non-colorability of certain finite unit-distance graphs with four or five colors.

Findings

Existence of finite unit-distance graphs not 4-colorable

Existence of finite unit-distance graphs not 5-colorable

New measure theoretic approach for geometric graph coloring

Abstract

A measure theoretic approach of the problem that there exits a finite unit-distance graphs in the plane that are not five (or four) colorable.