Monochromatic Translated Product and Answering Sahasrabudhe's Conjecture

This article resolves two related problems in Ramsey theory on the integers. We show that for any finite coloring of the set of natural numbers, there exist numbers $a$ and $b$ for which the configuration ${a, b, ab, a(b+1)}$ is monochromatic. By redefining the variables $a=x$ and $ab=y,$ our configurations transforms into ${x,y,x+y,\frac{y}{x}}.$ This finding has two main consequences: first, it disproves a conjecture proposed by J. Sahasrabudhe; second, it establishes a quotient version of the long-standing Hindman’s conjecture, which asks for a monochromatic set of the form ${x,y,x+y,xy}$.

💡 Research Summary

The paper addresses a long‑standing question in arithmetic Ramsey theory: whether certain mixed additive‑multiplicative configurations must appear monochromatically in any finite coloring of the natural numbers. Classical results guarantee that the additive Schur pattern ({a,b,a+b}) and its multiplicative analogue ({a,b,ab}) are partition‑regular. The natural next step is to ask whether the four‑element pattern ({a,b,a+b,ab}) is also partition‑regular. Hindman proved this for two‑colorings in 1979, and subsequently conjectured that it holds for any finite number of colors (the “Hindman conjecture”). The strongest partial result to date is due to Moreira (2017), who showed that ({a,ab,a+b}) is always monochromatic, using van der Waerden’s theorem and the theory of IP‑sets.

In a different direction, Sahasrabudhe (2017) conjectured that the asymmetric pattern ({a,b,a(b+1)}) could be avoided by a suitable coloring, suggesting that the “translated product” might not be partition‑regular.

The present work disproves Sahasrabudhe’s conjecture and, more strongly, proves that for every finite coloring of (\mathbb N) there exist natural numbers (a) and (b) such that the four‑element set
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