Monochromatic infinite sumsets

TL;DR

The paper constructs a specific rational vector space of large dimension where, under finite coloring, an infinite set exists with a monochromatic sumset, extending known results and establishing optimality under GCH.

Contribution

It demonstrates the existence of a large rational vector space with a monochromatic sumset property, filling a gap in the understanding of sumsets in high-dimensional spaces.

Findings

Existence of a rational vector space with large dimension where monochromatic sumsets occur

Extension of previous results to higher dimensions under GCH

Optimality of the result for the given dimension

Abstract

We show that there is a rational vector space $V$ such that, whenever $V$ is finitely coloured, there is an infinite set $X$ whose sumset $X+X$ is monochromatic. Our example is the rational vector space of dimension $\sup\{\aleph_0,2^{\aleph_0},2^{2^{\aleph_0}},\ldots\,\}$. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below $\aleph_\omega$. So our result is best possible under GCH.