Appears in collection : 2018 - T1 - WS1 - Model theory and combinatorics

In Ramsey Theory, ultrafilters often play an instrumental role. By means of nonstandard models, one can reduce those third-order objects (ultrafilters are sets of sets of natural numbers) to simple points. In this talk we present a nonstandard technique that is grounded on the above observation, and show its use to prove new results in Ramsey Theory of Diophantine equations. In this field, one investigates whether a given equation is partition regular on N (that is, there exist monochromatic solutions in every finite coloring of N).