Matthew Bowen

Monochromatic sums, products, and exponents in 2-colorings of the naturals.

We show for any $k\in \mathbb{N}$ that any $2$-coloring of $\mathbb{N}$ contains monochromatic sets of the form $\{x,y,xy,x+iy: i is less than k\}$ and $\{a,b,a^b,ab^i: i is less than k\}$. Previously, these were both only known in the special case $k=1$. Along the way we prove an exponential version of Moreira's theorem that holds for any number of colors and discuss sufficient conditions for a subset of $\mathbb{N}$ to contain patterns of these types. This is based on upcoming work with Alweiss and Sabok.