Monochromatic sumset without large cardinals

Abstract

We show in this note that in the forcing extension by Add(ω, ω ), the following Ramsey property holds: for any r ∈ ω and any f : R → r, there exists an infinite X ⊂ R such that X + X is monochromatic under f . We also show the Ramsey statement above is true in ZFC when r = 2. This answers two questions from [8].