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 ₘ denotes m collinear points with consecutive points of distance one apart, we say that Eⁿ → (ᵣ,ₛ) if there is a red/blue coloring of n-dimensional Euclidean space that avoids red congruent copies of ᵣ and blue congruent copies of ₛ. We show that Eⁿ → (₃, ₂₀), improving the best-known result Eⁿ → (₃, ₁₁₇₇) by F\"uhrer and T\'oth, and also establish Eⁿ → (₄, ₁₈) and Eⁿ → (₅, ₁₀) in the spirit of the classical result Eⁿ → (₆, ₆) due to Erd{{o}}s et. al. We also show a number of similar $3$-coloring results, as well as Eⁿ → (₃, α₆₈₈₉), where α is an arbitrary positive real number. This final result answers a question of F\"uhrer and T\'oth in the positive.