Almost-monochromatic sets and the chromatic number of the plane

TL;DR

This paper investigates the existence of almost monochromatic sets in various colourings of Euclidean and integer spaces, providing characterizations and proposing methods that could impact understanding the chromatic number of the plane.

Contribution

It characterizes almost monochromatic sets in integer and Euclidean spaces and suggests a novel approach towards proving the lower bound of the plane's chromatic number.

Findings

Characterization of almost monochromatic sets in $ ext{Z}$ and $ ext{R}^d$

Conditions under which such sets must exist in certain colourings

Proposed approach to potentially prove $oxed{ ext{chromatic number of } ext{R}^2 ext{ is at least 5}$

Abstract

In a colouring of $\mathbb{R}^d$ a pair $(S,s_0)$ with $S\subseteq \mathbb{R}^d$ and with $s_0\in S$ is \emph{almost monochromatic} if $S\setminus \{s_0\}$ is monochromatic but $S$ is not. We consider questions about finding almost monochromatic similar copies of pairs $(S,s_0)$ in colourings of $\mathbb{R}^d$, $\mathbb{Z}^d$, and in $\mathbb{Q}$ under some restrictions on the colouring. Among other results, we characterise those $(S,s_0)$ with $S\subseteq \mathbb{Z}$ for which every finite colouring of $\mathbb{R}$ without an infinite monochromatic arithmetic progression contains an almost monochromatic similar copy of $(S,s_0)$. We also show that if $S\subseteq \mathbb{Z}^d$ and $s_0$ is outside of the convex hull of $S\setminus \{s_0\}$, then every finite colouring of $\mathbb{R}^d$ without a similar monochromatic copy of $\mathbb{Z}^d$ contains an almost monochromatic similar copy of $(S,s_0)$. Further, we propose an approach of finding almost-monochromatic sets that might lead to a non-computer assisted proof of $\chi(\R^2)\geq 5$.