Prime and polynomial distances in colourings of the plane

TL;DR

This paper extends recent results on the chromatic number of the plane by showing that any finite coloring contains monochromatic pairs at distances given by polynomial values and prime numbers, highlighting new distance constraints.

Contribution

It introduces two new theorems: one about polynomial distances and another about prime distances in finite colorings of the plane, expanding the understanding of chromatic properties.

Findings

Monochromatic pairs at polynomial distances exist in any finite coloring.

Prime distances between monochromatic points are guaranteed in any finite coloring.

Extends previous work on the unbounded chromatic number of the odd distance graph.

Abstract

We give two extensions of the recent theorem of the first author that the odd distance graph has unbounded chromatic number. The first is that for any non-constant polynomial $f$ with integer coefficients and positive leading coefficient, every finite colouring of the plane contains a monochromatic pair of distinct points whose distance is equal to $f(n)$ for some integer $n$. The second is that for every finite colouring of the plane, there is a monochromatic pair of points whose distance is a prime number.