Lines in Euclidean Ramsey theory

Abstract

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 $m\ge 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erdős, 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 $K$.

7 pages