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=zn for arbitrary a∈Z+ and n≥2, and also of ax+by=aDzD+…+a1z∈Z[z] such that gcd(a,b)=1, D≥2, a,b,aD>0 and a1≠0.