Non-spherical sets versus lines in Euclidean Ramsey theory

TL;DR

This paper proves a conjecture in Euclidean Ramsey theory, showing that non-spherical sets can be avoided in certain colorings, while also controlling the existence of monochromatic progressions.

Contribution

It establishes a new result linking non-spherical sets and colorings in Euclidean spaces, confirming a conjecture by Wu and the first author.

Findings

Existence of colorings avoiding red copies of non-spherical sets

Construction of colorings avoiding long blue progressions

Verification of Wu and the first author's conjecture

Abstract

We show that for every non-spherical set $X$ in $\mathbb{E}^d$, there exists a natural number $m$ and a red/blue-colouring of $\mathbb{E}^n$ for every $n$ such that there is no red copy of X and no blue progression of length $m$ with each consecutive point at distance $1$. This verifies a conjecture of Wu and the first author.