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

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 \(a_1\neq0\).