Classification of partition-regular equations

Classify the equations over the positive integers that are partition regular under every finite coloring, thereby resolving the long-standing open problem of determining partition regularity for general equations in Ramsey theory.

Background

The paper defines partition regularity for equations over the positive integers as the property that every finite coloring admits a monochromatic solution. It notes that, despite major results such as Schur’s theorem and Rado’s characterization for linear systems, no general classification is known, particularly for nonlinear equations.