Prime and polynomial distances in colourings of the plane

2023/08/04 by James C. Davies, Davies, James, Rose McCarty +3 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications #Metric Geometry (math.MG) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2308.02483

openalex publication_date 2023/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Ramsey problems for graphs in Euclidean spaces and Cartesian powers

Related

• Odd distances in colourings of the plane
• Chromatic properties of the Euclidean plane
• The chromatic number of the plane is at least 5
• Coloring distance graphs on the plane
• The chromatic number of the plane with an interval of forbidden distances is at least 7