Monochromatic Translated Product and Answering Sahasrabudhe's Conjecture

TL;DR

This paper proves the existence of monochromatic configurations in finite colorings of natural numbers, disproves Sahasrabudhe's conjecture, and advances the understanding of Hindman's conjecture in Ramsey theory.

Contribution

It introduces a new monochromatic configuration involving sums and ratios, disproves a prior conjecture, and establishes a quotient version of Hindman's conjecture.

Findings

Existence of monochromatic sets of the form {a, b, ab, a(b+1)} in finite colorings.

Disproof of Sahasrabudhe's conjecture.

Establishment of a quotient version of Hindman's conjecture.

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