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We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if β denotes m collinear points with consecutive points of distance one apart, we say that EβΏ (α΅£, β) if there is a red/blue coloring of n-dimensional Euclidean space that avoids red congruent copies of α΅£ and blue congruent copies of β. We show that EβΏ (β, ββ), improving the best-known result EβΏ (β, ββββ) by F\"uhrer and T\'oth, and also establish EβΏ (β, ββ) and EβΏ (β , ββ) in the spirit of the classical result EβΏ (β, β) due to Erdos et. al. We also show a number of similar 3-coloring results, as well as EβΏ (β, ββββ), where is an arbitrary positive real number. This final result answers a question of F\"uhrer and T\'oth in the positive.
Currier et al. (Mon,) studied this question.
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