Sums, products, and exponents in two-colorings of the naturals

Ryan Alweiss, Matthew Bowen, Marcin Sabok

We prove that for any coloring of the naturals using two colors there are monochromatic sets of the form $\{x,y,xy,x+iy:i\le k\}$ and $\{x,y,x^y,x{y}^i:i\le k\}$ for any $k$.