Avoiding short progressions in Euclidean Ramsey theory II

Recently, Currier, Moore, and Yip developed a general framework for constructing colorings avoiding prescribed monochromatic arithmetic progressions in Euclidean Ramsey theory. Writing ₘ for an m-term arithmetic progression with unit spacing, they proved Eⁿ→(₃,₂₀) and Eⁿ→(₃,α₆₈₈₉) for every $α>0$. We refine and simplify their framework, proving Eⁿ→(₃,α₂₂) for every $α>0$. We further establish Eⁿ→(₃,α₂₁) except possibly for four values of α², and Eⁿ→(₃,α₁₇) for all but finitely many $α>0$. Our approach also recovers several earlier results and avoids some technical arguments used in previous works.