Monochromatic Translated Product and Answering Sahasrabudhe's Conjecture

Sayan Goswami

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arXiv · 2412.17868

Monochromatic Translated Product and Answering Sahasrabudhe's Conjecture

Sayan Goswami

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Abstract

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 $\{x,y,x+y,xy\}$.

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Sayan Goswami. 2024-12-21. Monochromatic Translated Product and Answering Sahasrabudhe's Conjecture. https://arxiv.org/abs/2412.17868

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