An affirmative answer to Owings's sumset question

Abstract

We give an affirmative answer to Owings's sumset question: for any $2$-coloring of natural numbers, there is an infinite $B\subseteq \mathbb{N}$ such that $B+B$ is monochromatic. More generally, for every $m,\ell \in \mathbb{N}$ and every $2$-coloring of $\mathbb{N}$, there is an infinite $B\subseteq \mathbb{N}$ such that $$(m+\ell )B\cup \{mx+\ell y:x,y\in B,x<y\}$$ is monochromatic.

In the newest version, we resolve the weighted form of Owings's sumset question completely