A nonstandard technique in combinatorial number theory

In Di Nasso (2015) and Luperi Baglini (2012) it has been introduced a technique, based on nonstandard analysis, to study some problems in combinatorial number theory. In this paper we review such a technique and we present three of its applications: the first one is a new proof of a known result regarding the algebra of βN, namely that the center of the semigroup (βN, ⊕) is N; the second one is a generalization of a theorem of Bergelson and Hindman on arithmetic progressions of length three; the third one regards the study of which polynomials in several variables with integers coefficients have a monochromatic solution for every finite coloring of N. We will study this last application in more detail: we will prove some algebraical properties of the set P of such polynomials and we will present a few examples of nonlinear polynomials in P. In the first part of the paper we will recall the main results of the nonstandard technique that we want to use, which is based on a characterization of ultrafilters by means of nonstandard analysis.