Density bounds for permutations avoiding monotone arithmetic progressions

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 \[ α_{\mathbb{N}}(3)\geq\frac23,\qquad α_{\mathbb{Z}}(3)\geq\frac23. \] We also prove ${\beta }_{\mathbb{Z}}(4)=1$ by constructing four-permutable subsets of the integers whose lower symmetric densities approach one.