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- W2624460271 abstract "A $textit{ladder}$ is a set $S subseteq mathbb Z^+$ such that any finite coloring of $mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $mathbb Z^+$ itself is a ladder. We also discuss variants of ladders, namely $textit{accessible}$ and $textit{walkable}$ sets, which are sets $S$ such that any coloring of $mathbb Z$ contains arbitrarily long (for accessible sets) or infinite (for walkable sets) monochromatic sequences with consecutive differences in $S$. We show that sets with upper density 1 are ladders and walkable. We also show that all directed graphs with infinite chromatic number are accessible, and reduce the bound on the walkability order of sparse sets from 3 to 2, making it tight." @default.
- W2624460271 created "2017-06-15" @default.
- W2624460271 creator A5061144291 @default.
- W2624460271 date "2017-06-06" @default.
- W2624460271 modified "2023-09-27" @default.
- W2624460271 title "Progressions and Paths in Colorings of $mathbb Z$" @default.
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