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

Down the Large Rabbit Hole