Down the Large Rabbit Hole

TL;DR

This paper explores the 2-Large Conjecture in Ramsey theory, aiming to understand if properties of monochromatic arithmetic progressions extend from two-colorings to multiple colors, and offers new insights and a research roadmap.

Contribution

It provides new results and a strategic roadmap for investigating the 2-Large Conjecture in Ramsey theory on integers.

Findings

New partial results related to the 2-Large Conjecture

A proposed framework for future research in the area

Insights into the extension of monochromatic progression properties

Abstract

This article documents my journey down the rabbit hole, chasing what I have come to know as a particularly unyielding problem in Ramsey theory on the integers: the $2$-Large Conjecture. This conjecture states that if $D \subseteq \mathbb{Z}^+$ has the property that every $2$-coloring of $\mathbb{Z}^+$ admits arbitrarily long monochromatic arithmetic progressions with common difference from $D$ then the same property holds for any finite number of colors. We hope to provide a roadmap for future researchers and also provide some new results related to the $2$-Large Conjecture.