Monochromatic products and sums in 2-colorings of N

Abstract

We show that any 2-coloring of $N$ contains infinitely many monochromatic sets of the form $\{x,y,xy,x+y\}$, and more generally monochromatic sets of the form $\{x_i,\prod x_i,\sum x_i:i\le n\}$ for any $n\in N$. Along the way we prove a monochromatic products of sums theorem that extends Hindman's theorem and a colorful variant of this result that holds in any ‘balanced’ coloring.