论文标题
旨在表征表格$ ax+by = p（z）$的2•ramsey方程
Towards characterizing the 2-Ramsey equations of the form $ax+by=p(z)$
论文作者
论文摘要
在本文中，我们研究了$ ax+by = p（z）$的等式的拉姆齐型问题。我们表明，如果某些技术假设成立，那么积极整数的任何2色就可以无限地接受方程式$ ax+by = p（z）$的许多单色解决方案。这需要几种值得注意的情况的$ 2 $ -RAMSEYNESS，例如$ ax+y = y = z^n $，用于任意$ a \ in \ mathbb {z}^{+} $和$ n \ ge 2 $，以及$ ax+by的$ ax+by = a_dz^d+\ d+\ \ d+\ dots+a_1z $ a _1z] $ \ text {gcd}（a，b）= 1 $，$ d \ ge 2 $，$ a，b，a_d> 0 $和$ a_1 \ neq0 $。
In this paper, we study a Ramsey-type problem for equations of the form $ax+by=p(z)$. We show that if certain technical assumptions hold, then any 2-colouring of the positive integers admits infinitely many monochromatic solutions to the equation $ax+by=p(z)$. This entails the $2$-Ramseyness of several notable cases such as the equation $ax+y=z^n$ for arbitrary $a\in\mathbb{Z}^{+}$ and $n\ge 2$, and also of $ax+by=a_Dz^D+\dots+a_1z\in\mathbb{Z}[z]$ such that $\text{gcd}(a,b)=1$, $D\ge 2$, $a,b,a_D>0$ and $a_1\neq0$.
