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.
Categories: math.CO math.NT