More on lines in Euclidean Ramsey theory

In brief

• Question. Can Euclidean space be two-coloured, in every dimension, with no red copy of three equally spaced points on a line and no blue copy of some longer such configuration?
• Result. Yes: for some natural number m there is a red/blue colouring of Euclidean space in every dimension with no red copy of l_3 and no blue copy of l_m, answering a question of Conlon and Fox, and independently of Arman and Tsaturian.

Abstract

Let l_m be a sequence of m points on a line with consecutive points at distance one. Answering a question raised by Fox and the first author and independently by Arman and Tsaturian, we show that there is a natural number m and a red/blue-colouring of En for every n that contains no red copy of l_3 and no blue copy of l_m.

Cite

David Conlon and Yu-Han Wu (2023). More on lines in Euclidean Ramsey theory. Comptes Rendus. Mathématique, 361(G5), 897–901.

@article{conlon2023more,
  title={More on lines in {Euclidean} {Ramsey} theory},
  author={Conlon, David and Wu, Yu-Han},
  journal={Comptes Rendus. Math{\'e}matique},
  volume={361},
  number={G5},
  pages={897--901},
  year={2023}
}