Prime and polynomial distances in colourings of the plane

Autor:	Davies, James, McCarty, Rose, Pilipczuk, Michał
Rok vydání:	2023
Předmět:	Mathematics - Combinatorics Mathematics - Metric Geometry Mathematics - Number Theory
Druh dokumentu:	Working Paper
Popis:	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. Comment: 24 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2308.02483 Zobrazit plný text záznamu View this record from Arxiv