Monochromatic Translated Product and Answering Sahasrabudhe's Conjecture
Sayan Goswami
TL;DR
The paper addresses the problem of finding monochromatic configurations that fuse additive and multiplicative structure in finite colorings of $\mathbb{N}$. It employs IP-set methods, Hindman’s theorem, and van der Waerden’s theorem (building on Moreira’s work) to construct color-persistent configurations. The main result shows that for every finite coloring of $\mathbb{N}$ there exist $a,b$ with the monochromatic set $\{a,b,ab,(a+1)b\}$, thereby disproving Sahasrabudhe's conjecture and yielding a Hindman-type variant with $\{a,b,ab,a(b+1)\}$. This advances the understanding of sum-product phenomena in Ramsey theory and provides new tools for studying partition-regular configurations on the integers.
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 $\{a, b, ab, (a+1)b\}$ is monochromatic. This finding has two main consequences: first, it disproves a conjecture proposed by J. Sahasrabudhe; second, it establishes a variant of the long-standing Hindman's conjecture, which asks for a monochromatic set of the form $\{a, b, ab, a+b\}$.