Monochromatic Sums and Products of Polynomials

TL;DR

This paper proves that the pattern x, x+y, xy is partition regular over formal integer polynomials with primitive recursive bounds, offering a new proof and the first primitive recursive bound for x, x+y, xy over natural numbers.

Contribution

It introduces a novel proof for the partition regularity of x, x+y, xy over natural numbers with primitive recursive bounds, extending to formal polynomials.

Findings

Partition regularity of x, x+y, xy over formal integer polynomials.

First primitive recursive bound for the pattern over x, x+y, xy.

New proof technique for partition regularity results.

Abstract

We show that the pattern $\{x,x+y,xy\}$ is partition regular over the space of formal integer polynomials of degree at least one with zero constant term, with primitive recursive bounds. This provides a new proof for the partition regularity of $\{x,x+y,xy\}$ over $\mathbb{N}$, which gives the first primitive recursive bound.