31 research outputs found
Partition Regularity of Nonlinear Polynomials: a Nonstandard Approach
In 2011, Neil Hindman proved that for every natural number the
polynomial \begin{equation*} \sum_{i=1}^{n} x_{i}-\prod\limits_{j=1}^{m} y_{j}
\end{equation*} has monochromatic solutions for every finite coloration of
. We want to generalize this result to two classes of nonlinear
polynomials. The first class consists of polynomials
of the following kind: \begin{equation*}
P(x_{1},...,x_{n},y_{1},...,y_{m})=\sum_{i=1}^{n}a_{i}x_{i}M_{i}(y_{1},...,y_{m}),
\end{equation*} where are natural numbers,
has monochromatic solutions for every finite
coloration of and the degree of each variable in
is at most one. An example of such a polynomial is
\begin{equation*} x_{1}y_{1}+x_{2}y_{1}y_{2}-x_{3}.\end{equation*} The second
class of polynomials generalizing Hindman's result is more complicated to
describe; its particularity is that the degree of some of the involved
variables can be greater than one.\\ The technique that we use relies on an
approach to ultrafilters based on Nonstandard Analysis. Perhaps, the most
interesting aspect of this technique is that, by carefully chosing the
appropriate nonstandard setting, the proof of the main results can be obtained
by very simple algebraic considerations
A nonstandard technique in combinatorial number theory
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
Ultrafilters maximal for finite embeddability
In [1] the authors showed some basic properties of a pre-order that arose in
combinatorial number theory, namely the finite embeddability between sets of
natural numbers, and they presented its generalization to ultrafilters, which
is related to the algebraical and topological structure of the Stone-\v{C}ech
compactification of the discrete space of natural numbers. In this present
paper we continue the study of these pre-orders. In particular, we prove that
there exist ultrafilters maximal for finite embeddability, and we show that the
set of such ultrafilters is the closure of the minimal bilateral ideal in the
semigroup (\bN,\oplus), namely \overline{K(\bN,\oplus)}. As a consequence,
we easily derive many combinatorial properties of ultrafilters in
\overline{K(\bN,\oplus)}. We also give an alternative proof of our main
result based on nonstandard models of arithmetic
Un nuovo modo di contare l'infinito
Si definisce un metodo, basato sugli ultrafiltri,
per definire la grandezza di opportune classi di insiemi in modo che gli insiemi di grandezza soddisfino sia determinate proprietà algebriche che i principi di Euclide per grandezze