Monochromatic quotients, products and polynomial sums in the rationals

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 $$\{x,\frac{x}{y^a},x+p_1(y),\cdots ,x+p_k(y)\}$$ and there are non-zero $v,u\in \mathbb{Q}$ such that there exists a color which contains a set of the form $$\{v,v\cdot u^a,v+p_1(u),\cdots ,v+p_k(u)\}.$$

References in corpus (1)

• On the interplay between notions of additive and multiplicative largeness and its combinatorial applications