Monochromatic Sums and Quotients Near Zero

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.

10 pages