More on lines in Euclidean Ramsey theory

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 $\ell_m$.