Avoiding short progressions in Euclidean Ramsey theory

TL;DR

This paper develops a framework for constructing colorings in Euclidean space that avoid short monochromatic arithmetic progressions, advancing known bounds and answering open questions in Euclidean Ramsey theory.

Contribution

It introduces a general method to avoid short progressions in Euclidean Ramsey problems and improves existing bounds, including a positive answer to a previously open question.

Findings

Established new bounds for avoiding short progressions in Euclidean space.

Improved the known result from fcrer and Tf3th's work.

Provided multiple new coloring constructions 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 $\mathbb{E}^n \not \to (\ell_r,\ell_s)$ 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 $\mathbb{E}^n \not \to (\ell_3, \ell_{20})$, improving the best-known result $\mathbb{E}^n \not \to (\ell_3, \ell_{1177})$ by F\"uhrer and T\'oth, and also establish $\mathbb{E}^n \not \to (\ell_4, \ell_{14})$ and $\mathbb{E}^n \not \to (\ell_5, \ell_{8})$ in the spirit of the classical result $\mathbb{E}^n \not \to (\ell_6, \ell_{6})$ due to Erd\H{o}s et. al. We also show a number of similar $3$-coloring results, as well as $\mathbb{E}^n \not \to (\ell_3, \alpha\ell_{6889})$, where $\alpha$ is an arbitrary positive real number. This final result answers a question of F\"uhrer and T\'oth in the positive.