Towards characterizing the 2-Ramsey equations of the form $ax+by=p(z)$

Abstract

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 $ax+by=a_D{z}^D+\cdots +a_{1}z\in \mathbb{Z}[z]$ such that $\text{gcd}(a,b)=1$, $D\ge 2$, $a,b,a_D>0$ and $a_1\ne 0$.

16 pages