Mathematics > Combinatorics
[Submitted on 30 Apr 2024]
Title:Avoiding short progressions in Euclidean Ramsey theory
View PDF HTML (experimental)Abstract: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 Führer 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 Führer and Tóth in the positive.
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