Monochromatic paths for the integers

TL;DR

This paper investigates the properties of differences in monochromatic arithmetic sequences within finite colorings of the natural numbers, extending van der Waerden's theorem to new aspects.

Contribution

It introduces new insights into the set of differences in monochromatic sequences, expanding understanding beyond traditional arithmetic progression results.

Findings

Characterization of difference sets in monochromatic sequences

Conditions for the existence of certain difference patterns

Extensions of van der Waerden's theorem to difference sets

Abstract

Recall that van der Waerden's theorem states that any finite coloring of the naturals has arbitrarily long monochromatic arithmetic sequences. We explore questions about the set of differences of those sequences.