Year: 2017
Type: preprint
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 $\aleph_ω$. So our result is best possible under GCH.
Source: arXiv (Cornell University)
Authors: Leader, Imre, Russell, Paul A.
Institution: University of Cambridge
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2737718316