Monochromatic products and sums in $2$-colorings of $\mathbb{N}$ (2205.12921v1)

Abstract: We show that any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic sets of the form ${x,y,xy,x+y},$ and more generally monochromatic sets of the form ${x_i,\prod x_i,\sum x_i: i\leq k}$ for any $k\in\mathbb{N}.$ 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.