# Progressions and Paths in Colorings of $\mathbb Z$

## TL;DR

This paper explores various types of sets in the integers that guarantee long monochromatic progressions under finite colorings, extending classical results like Van der Waerden's theorem to new classes of sets.

## Contribution

It introduces and analyzes accessible and walkable sets, establishing their properties and relationships, including density conditions and graph-theoretic implications.

## Key findings

- Sets with density 1 are ladders and walkable.
- All directed graphs with infinite chromatic number are accessible.
- Reduced the walkability order bound from 3 to 2 for sparse sets.

## 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.