论文标题
理性中的单色商，产品和多项式总和
Monochromatic quotients, products and polynomial sums in the rationals
论文作者
论文摘要
令$ k，a \ in \ mathbb {n} $，让$ p_1，\ cdots，p_k \ in \ mathbb {q} [q} [n] $，零常数项。我们表明，对于$ \ mathbb {q} $的任何有限颜色，都有non-Zero $ x，y \ in \ mathbb {q} $，因此存在包含一组表单的颜色$ \ big \ {x，\ frac {x} {y^a}，x+p_ {1}（y），\ cdots，x+p_ {k {k}（y）\ big \} $$，并且非零v，in \ Zero $ v，in \ n y Mathbb {Q} $ $$ \ big \ {v，v \ cdot {u^a}，v+p_ {1}（u），\ cdots，v+p_ {k}（k}（u）\ big \}。$$
Let $k,a\in \mathbb{N}$ and let $p_1,\cdots,p_k\in \mathbb{Q}[n]$ with zero constant term. We show that for any finite coloring of $\mathbb{Q}$, there are non-zero $x,y\in \mathbb{Q}$ such that there exists a color which contains a set of the form $$\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\}.$$
