Infinite monochromatic sumsets for colourings of the reals

TL;DR

This paper demonstrates that under certain set-theoretic assumptions, any finite coloring of the real numbers admits an infinite sumset that is monochromatic, contrasting previous results about the non-existence of such sets.

Contribution

It proves a consistency result showing the existence of monochromatic infinite sumsets for any finite coloring of the reals under specific set-theoretic assumptions.

Findings

Existence of monochromatic infinite sumsets under certain assumptions

Contrasts with prior results on finite colorings of reals

Provides a set-theoretic consistency proof

Abstract

N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $\mathbb R$ so that no infinite sumset $X+X=\{x+y:x,y\in X\}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:\mathbb R\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb R$ so that $c$ is constant on $X+X$.