English

Non-spherical sets versus lines in Euclidean Ramsey theory

Combinatorics 2025-09-10 v1

Abstract

We show that for every non-spherical set XX in Ed\mathbb{E}^d, there exists a natural number mm and a red/blue-colouring of En\mathbb{E}^n for every nn such that there is no red copy of X and no blue progression of length mm with each consecutive point at distance 11. This verifies a conjecture of Wu and the first author.

Keywords

Cite

@article{arxiv.2406.07718,
  title  = {Non-spherical sets versus lines in Euclidean Ramsey theory},
  author = {David Conlon and Jakob Führer},
  journal= {arXiv preprint arXiv:2406.07718},
  year   = {2025}
}
R2 v1 2026-06-28T17:02:20.349Z