  Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of
distance one. For every natural number $n$, we prove the existence of a
red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no
blue copy of $\ell_m$ for any $m \geq 2^{cn}$. This is best possible up to the
constant $c$ in the exponent. It also answers a question of Erd\H{o}s, Graham,
Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every
natural number $n$, there is a set $K \subset \mathbb{E}^1$ and a
red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no
blue copy of $K$.
