Nonstandard methods for Ramsey theory

It is no coincidence that methods based on the use of infinitesimal or infinite quantities are usually referred to as "non-Archimedean". In fact, in his "The Method of Mechanical Theorems" the genius Greek mathematician used such quantities to calculate areas and volumes of many geometrical figures. Infinitesimal were also one of the key concepts used by Leibniz in the latter part of the seventeenth century to discover differential and integral calculus; however, due mainly to some foundational problems, for nearly two centuries the use of non-Archimedean methods almost vanished from mathematics (with some notable exceptions, including Du Bois-Reymond, Hilbert and Levi Civita). The situation changed in 1966, when Robinson discovered nonstandard analysis, solving the foundational issues of non-Archimedean mathematics. Since then, the relevance of non-Archimedean methods across mathematics has risen again, and in many cases nonstandard analysis has played a key role in such developments. "Complete chaos is impossible" is a catching way of expressing the aims of Ramsey theory: usually, results in this area express the fact that large enough sets must contain organized structures. Many problems in mathematics, logic and informatics can be framed in this way. In this context, interesting open problems often regard structures arising from nonlinear equations. Due to some recent developments, the interest of the mathematical community towards such problems has grown, and we might be on the verge of obtaining a significant breakthrough. Achieving this breakthrough by means of a combination of standard and nonstandard techniques is the main goal of this project.