Avoiding short progressions in Euclidean Ramsey theory
Gabriel Currier, Kenneth Moore, Chi Hoi Yip
TL;DR
The paper advances Euclidean Ramsey theory by introducing a general modular-square-norm coloring framework that produces strong negative results for short monochromatic progressions. It defines red points as those with $\lfloor d|x|^2\rfloor \in S \pmod p$ and uses corollaries and a finite-checking proposition to certify the absence of red $\ell_r$ while prohibiting blue $\ell_s$ for large $s$, enabling explicit parameter constructions. The main contributions include $\mathbb{E}^n \not\to (\ell_3,\ell_{20})$, $\mathbb{E}^n \not\to (\ell_4,\ell_{14})$, $\mathbb{E}^n \not\to (\ell_5,\ell_{8})$, and a uniform result $\mathbb{E}^n \not\to (\ell_3, \alpha\ell_{6889})$ for all $\alpha>0$, along with multi-coloring extensions and parallelogram-family results. The approach combines explicit parameter choices, corollaries that constrain possible progressions, and computational verification, yielding a versatile method that extends beyond spherical colorings and answers questions posed by Führer and Tóth. Overall, the framework provides a concrete, scalable toolkit for constructing colorings that avoid a broad class of short monochromatic configurations in Euclidean spaces.
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 Führer and Tóth, 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 Erdős et. al. We also show a number of similar $3$-coloring results, as well as $\mathbb{E}^n \not \to (\ell_3, α\ell_{6889})$, where $α$ is an arbitrary positive real number. This final result answers a question of Führer and Tóth in the positive.