论文标题
规范的多项式范德沃登的定理
A Canonical Polynomial Van der Waerden's Theorem
论文作者
论文摘要
我们证明了一个规范的多项式Van der Waerden的定理。更确切地说，我们显示以下内容。令$ \ {p_1（x），\ ldots，p_k（x）\} $为一组多项式，以使$ p_i（x）\ in \ mathbb {z} [x] $ and $ p_i（0）= 0 $，对于每个$ i \ in \ in \ in \ in \ {1，\ ldots，k \ ldots，k \} $。然后，在$ \ mathbb {z} $的任何着色中，存在$ a，d \ in \ mathbb {z} $，使得$ \ {a+p_1（d），\ ldots，a+p_ {k}（k}（d）\} $单色或彩虹或彩虹集。
We prove a canonical polynomial Van der Waerden's Theorem. More precisely, we show the following. Let $\{p_1(x),\ldots,p_k(x)\}$ be a set of polynomials such that $p_i(x)\in \mathbb{Z}[x]$ and $p_i(0)=0$, for every $i\in \{1,\ldots,k\}$. Then, in any colouring of $\mathbb{Z}$, there exist $a,d\in \mathbb{Z}$ such that $\{a+p_1(d),\ldots,a+p_{k}(d)\}$ forms either a monochromatic or a rainbow set.
