A Finite Graph Approach to the Probabilistic Hadwiger-Nelson Problem

TL;DR

This paper introduces a probabilistic framework linking finite unit-distance graphs to the Hadwiger-Nelson problem, providing improved bounds on graph colorability and advancing understanding of plane coloring complexities.

Contribution

It develops a novel probabilistic approach connecting finite graphs to the plane coloring problem, improving bounds for non-colorable graphs and exploring a probabilistic version of the de Bruijn-Erdős theorem.

Findings

Lower bounds on badness for k-colorings of the plane.

Improved bounds on non k-colorable unit-distance graphs.

Partial progress on a probabilistic de Bruijn-Erdős theorem.

Abstract

We advance a probabilistic approach to the Hadwiger-Nelson problem initially developed by the Polymath16 project, in particular relating the approach to finite unit-distance graphs. We define the numerical \textit{badness} of a given $k$-coloring of the plane to be the probability that a randomly chosen unit-distance edge is monochromatic under the coloring, and we provide lower bounds on the badness of arbitrary $k$-colorings using a probabilistic technique relating to finite graphs. The contrapositive of the resulting bounds lets us compute lower bounds on the order of non $k$-colorable unit-distance graphs, improving bounds produced by Pritikin and the Polymath16 project in the $k = 4$ and $k = 5$ cases. Additionally, we make partial progress on a probabilistic analog of the de Bruijn-Erd\H{o}s compactness theorem.