Avoiding short progressions in Euclidean Ramsey theory

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 Tóth, and also establish \(\mathbb{E}^n \not \to (\ell_4, \ell_{18})\) and \(\mathbb{E}^n \not \to (\ell_5, \ell_{10})\) in the spirit of the classical result \(\mathbb{E}^n \not \to (\ell_6, \ell_{6})\) due to Erd{ő}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 Tóth in the positive.