Monochromatic sumset without large cardinals
Volume 250 / 2020
Fundamenta Mathematicae 250 (2020), 243-252
MSC: Primary 03E02, 03E35, 03E55.
DOI: 10.4064/fm841-12-2019
Published online: 13 March 2020
Abstract
We show in this note that in the forcing extension by $\mathrm {Add}(\omega ,\beth _{\omega })$, the following Ramsey property holds: for any $r\in \omega $ and any $f: \mathbb {R}\to r$, there exists an infinite $X\subset \mathbb {R}$ such that $X+X$ is monochromatic under $f$. We also show the Ramsey statement above is true in $\mathrm {ZFC}$ when $r=2$. This answers two questions of Komjáth et al. (2019).
