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 β N , namely that the center of the semigroup ( β N , ⊕ ) 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. [ABSTRACT FROM AUTHOR]