A limiting result for the Ramsey theory of functional equations

We study systems of functional equations whose solutions can be parameterized by one of the variables. Our main result proves that the partition regularity (PR) of such systems can be completely characterized by the existence of constant solutions. As applications of this result, we prove the following:

$\bullet$ A complete characterization of PR systems of Diophantine equations in two variables over $\mathbb N$. In particular, we prove that the only infinitely PR irreducible equation in two variables is $x=y$.

$\bullet$ PR of $S$-unit equations and the failure of Rado’s Theorem for finitely generated multiplicative subgroups of $\mathbb C$.

$\bullet$ A complete characterization of the PR of two classes of polynomial exponential equations.