Monochromatic quotients, products and polynomial sums in the rationals

TL;DR

This paper proves that in any finite coloring of the rational numbers, certain monochromatic configurations involving polynomial sums and multiplicative structures always exist, extending classical combinatorial results.

Contribution

It introduces new combinatorial structures in the rationals involving polynomial sums and products, demonstrating their unavoidable monochromatic presence under any finite coloring.

Findings

Existence of monochromatic polynomial sum configurations in rationals.

Existence of monochromatic multiplicative configurations in rationals.

Extension of classical Ramsey theory to polynomial and multiplicative structures in Q.

Abstract

Let $k,a\in \mathbb{N}$ and let $p_1,\cdots,p_k\in \mathbb{Q}[n]$ with zero constant term. We show that for any finite coloring of $\mathbb{Q}$, there are non-zero $x,y\in \mathbb{Q}$ such that there exists a color which contains a set of the form $$\Big\{x,\frac{x}{y^a},x+p_{1}(y),\cdots,x+p_{k}(y)\Big\}$$ and there are non-zero $v,u\in \mathbb{Q}$ such that there exists a color which contains a set of the form $$\Big\{v,v\cdot {u^a},v+p_{1}(u),\cdots,v+p_{k}(u)\Big\}.$$