Avoiding short progressions in Euclidean Ramsey theory

Abstract

We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if ${\ell }_m$ denotes $m$ collinear points with consecutive points of distance one apart, we say that ${\mathbb{E}}^n\nrightarrow ({\ell }_r,{\ell }_s)$ if there is a red/blue coloring of $n$-dimensional Euclidean space that avoids red congruent copies of ${\ell }_r$ and blue congruent copies of ${\ell }_s$. We show that ${\mathbb{E}}^n\nrightarrow ({\ell }_3,{\ell }_{20})$, improving the best-known result ${\mathbb{E}}^n\nrightarrow ({\ell }_3,{\ell }_{1177})$ by Führer and Tóth, and also establish ${\mathbb{E}}^n\nrightarrow ({\ell }_4,{\ell }_{14})$ and ${\mathbb{E}}^n\nrightarrow ({\ell }_5,{\ell }_8)$ in the spirit of the classical result ${\mathbb{E}}^n\nrightarrow ({\ell }_6,{\ell }_6)$ due to Erdős et. al. We also show a number of similar $3$-coloring results, as well as ${\mathbb{E}}^n\nrightarrow ({\ell }_3,\hat{I}\pm {\ell }_{6889})$, where $\hat{I}\pm$ is an arbitrary positive real number. This final result answers a question of Führer and Tóth in the positive.

14 pages, revised based on referee comments