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

TL;DR

This paper investigates the 2-Ramsey property of equations of the form $ax+by=p(z)$, establishing conditions under which any 2-coloring of positive integers yields infinitely many monochromatic solutions, including notable polynomial cases.

Contribution

The paper characterizes conditions for the 2-Ramsey property of a broad class of equations, extending known results to polynomial and specific linear forms.

Findings

Infinitely many monochromatic solutions exist under certain assumptions.

Notable cases like $ax+y=z^n$ are 2-Ramsey.

Polynomial equations with specific gcd and degree conditions are 2-Ramsey.

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 $ax+by=a_Dz^D+\dots+a_1z\in\mathbb{Z}[z]$ such that $\text{gcd}(a,b)=1$, $D\ge 2$, $a,b,a_D>0$ and $a_1\neq0$.