More on lines in Euclidean Ramsey theory

Abstract

Let ${\ell }_m$ 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 ${\mathbb{E}}^n$ for every $n$ that contains no red copy of ${\ell }_3$ and no blue copy of ${\ell }_m$.

4 pages