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

TL;DR

This paper proves that in any two-coloring of natural numbers, certain structured monochromatic sets involving sums, products, and exponents always exist for any size parameter k.

Contribution

It establishes the existence of complex monochromatic configurations involving sums, products, and exponents in two-colorings of the natural numbers, extending previous combinatorial results.

Findings

Existence of monochromatic sets with sum and product structures

Presence of exponential configurations in two-colorings

Results hold for any size parameter k

Abstract

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