Ultrafilters, monads and combinatorial properties

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as by characterizing their combinatorial properties by means of their monads. This extends to arbitrary sets methods previously used to study partition regular Diophantine equations on \N. Several applications are described by means of multiple examples.