A probabilistic Hadwiger-Nelson problem

TL;DR

This paper explores the probabilistic coloring problem related to the Hadwiger-Nelson problem, analyzing how to maximize the probability that endpoints of a randomly thrown needle on a colored table fall on different colors, with implications for finite graphs with unit-length edges.

Contribution

It introduces a probabilistic framework for the Hadwiger-Nelson problem and derives bounds on the maximum probability for such colorings.

Findings

Bounds on the optimal probability for the coloring problem

Connection between probabilistic coloring and finite graphs with unit edges

Insights into the structure of colorings maximizing the probability

Abstract

If you color a table using k colors, and throw a needle randomly on it, for some proper definition, you get a certain probability that the endpoints will fall on different colors. How can one make this probability maximal? This problem is related to finite graphs having unit-length edges, and some bounds on the optimal probability are deduced.