Monochromatic Sums and Quotients Near Zero

TL;DR

The paper extends results on monochromatic sets involving sums and products to dense subsemigroups near zero, proving certain patterns are monochromatic or partition regular in this context.

Contribution

It proves that specific algebraic patterns are monochromatic or partition regular near zero in dense subsemigroups of the positive reals, extending previous finite coloring results.

Findings

The set {a, b, ab, b(a+1)} is monochromatic near zero in dense subsemigroups.

The pattern x, y, x+y, y/x is partition regular near zero.

Monochromaticity holds for these patterns under any finite coloring of dense subsemigroups.

Abstract

Recently S. Goswami proved that whenever the set $\mathbb N$ of natural numbers is finitely colored, the set $\{a, b, ab, b(a+1)\}$ is monochromatic which also established a variant of the long-standing Hindman's conjecture, which asks for a monochromatic set of the form $\{a, b, ab, a+b\}$. Actually he disproved a conjecture proposed by J. Sahasrabudhe that $\{a, b, a(b + 1)\}$ is not partition regular. In this paper we prove that $\{a, b, ab, b(a+1)\}$ is monochromatic near zero which means for every finite coloring of a dense subsemigroups of $((0, \infty), +)$, the set $\{a, b, ab, b(a+1)\}$ is monochromatic near zero or in other words, we will get $a, b$ in a dense subsemigroups of $((0, \infty), +)$ as small as we want such that the set $\{a, b, ab, b(a+1)\}$ is monochromatic for every finite coloring of that dense subsemigroups of $((0, \infty), +)$, also we show that the pattern $x, y, x+y, \frac{y}{x}$ is partition regular near zero.