Pairwise sums in colourings of the reals

TL;DR

This paper explores the limitations and possibilities of finding monochromatic sumset structures within finite colourings of the reals, demonstrating both existence and non-existence results under different set-theoretic assumptions.

Contribution

It establishes new results on the existence of monochromatic sumsets in colourings of the reals, contrasting the effects of measurability, Baire property, and the continuum hypothesis.

Findings

Existence of colourings with no uncountable monochromatic sumsets under CH.

Existence of colourings with no infinite monochromatic sumsets under CH.

Measurable or Baire colourings guarantee infinite or uncountable monochromatic sumsets.

Abstract

Suppose that we have a finite colouring of the reals. What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums monochromatic. We also show that there is such a colouring such that there is no infinite set X with X+X (the pairwise sums from X, allowing repetition) monochromatic. These results assume CH. In the other direction, we show that if each colour class is measurable, or each colour class is Baire, then there is an infinite set X (and even an uncountable X, of size the reals) with X+X monochromatic. We also give versions for all of these results for k-wise sums in place of pairwise sums.