1,860 research outputs found
Hypernatural Numbers as Ultrafilters
In this paper we present a use of nonstandard methods in the theory of
ultrafilters and in related applications to combinatorics of numbers
A taste of nonstandard methods in combinatorics of numbers
By presenting the proofs of a few sample results, we introduce the reader to the use of nonstandard analysis in aspects of combinatorics of numbers
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
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