Down the large rabbit hole

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.