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arXiv · 2608.12604

Density bounds for permutations avoiding monotone arithmetic progressions

Jesse Geneson
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Abstract

For $X\in\{\mathbb{N},\mathbb{Z}\}$, let $\alpha_X(\ell)$ and $\beta_X(\ell)$ denote the supremal upper and lower densities of subsets of $X$ admitting $\omega$-permutations without monotone $\ell$-term arithmetic progressions. We strengthen the published lower bounds for the three-term upper-density parameters by proving \[ \alpha_{\mathbb{N}}(3)\geq\frac23,\qquad \alpha_{\mathbb{Z}}(3)\geq\frac23. \] We also prove $\beta_{\mathbb{Z}}(4)=1$ by constructing four-permutable subsets of the integers whose lower symmetric densities approach one.

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Jesse Geneson. 2026-08-12. Density bounds for permutations avoiding monotone arithmetic progressions. https://arxiv.org/abs/2608.12604

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