  In [9], [15] it has been introduced a technique, based on nonstandard
analysis, to study some problems in combinatorial number theory. In this paper
we present three applications of this technique: the first one is a new proof
of a known result regarding the algebra of \betaN, namely that the center of
the semigroup (\beta\mathbb{N};\oplus) is \mathbb{N}; the second one is a
generalization of a theorem of Bergelson and Hindman on arithmetic progressions
of lenght three; the third one regards the partition regular polynomials in
Z[X], namely the polynomials in Z[X] that have a monochromatic solution for
every finite coloration of N. We will study this last application in more
detail: we will prove some algebraical properties of the sets of such
polynomials and we will present a few examples of nonlinear partition regular
polynomials. 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.
