Monochromatic Sumset Without the use of large cardinals

Abstract

We show in this note that in the forcing extension by $Add(\omega ,{\beth }_{\omega })$, 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 by Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky.