Partition Regularity of Nonlinear Polynomials: a Nonstandard Approach

TL;DR

This paper extends Hindman's partition regularity results to broader classes of nonlinear polynomials using a nonstandard analysis approach, simplifying proofs through algebraic considerations of ultrafilters.

Contribution

It generalizes partition regularity to new classes of nonlinear polynomials and introduces a nonstandard analysis technique that simplifies the proof process.

Findings

Partition regularity holds for certain nonlinear polynomials.

Nonstandard analysis provides a straightforward algebraic proof method.

The approach applies to polynomials with variables of degree at most one and some with higher degrees.

Abstract

In 2011, Neil Hindman proved that for every natural number $n,m$ 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 $\mathbb{N}$. We want to generalize this result to two classes of nonlinear polynomials. The first class consists of polynomials $P(x_{1},...,x_{n},y_{1},...,y_{m})$ 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 $n,m$ are natural numbers, $\sum\limits_{i=1}^{n}a_{i}x_{i}$ has monochromatic solutions for every finite coloration of $\mathbb{N}$ and the degree of each variable $y_{1},...,y_{m}$ in $M_{i}(y_{1},...,y_{m})$ 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.