Some New Results on Monochromatic Sums and Products in the Rationals

TL;DR

This paper demonstrates that for certain subsets of rationals and naturals, it is impossible to find infinite sets with all finite sums and products sharing the same color under finite colorings, revealing new combinatorial properties.

Contribution

It introduces new finite colorings of rationals and naturals that prevent infinite monochromatic sum-product sets, extending previous results with a uniform approach.

Findings

Finite coloring of rationals with restricted denominators avoids infinite monochromatic sum-product sets

A new short proof for the coloring of naturals preventing infinite monochromatic pairwise sums and products

Generalization of results to rationals with denominators containing only finitely many primes

Abstract

Our aim in this paper is to show that, for any $k$, there is a finite colouring of the set of rationals whose denominators contain only the first $k$ primes such that no infinite set has all of its finite sums and products monochromatic. We actually prove a `uniform' form of this: there is a finite colouring of the rationals with the property that no infinite set whose denominators contain only finitely many primes has all of its finite sums and products monochromatic. We also give various other results, including a new short proof of the old result that there is a finite colouring of the naturals such that no infinite set has all of its pairwise sums and products monochromatic.