Discrete Approximation of Coloring the Plane

Abstract

 This work is inspired by the open problem The Chromatic Number of The Plane. We approximate the plane by only considering points with coordinates (p/n, q/n), where p and q are integers, and n is a fixed positive integer. To prevent triviality, we add the restriction that two points are adjacent if and only if their distance is between 1-ε and 1+ε for a non-negative ε. Adjacent points have to be colored differently. Our goal is to find the smallest ε for each n that will force us to use five colors.