Monochromatic Sums and Products with Additive or Multiplicative Shifts in Natural Numbers

Abstract

In this paper, we prove that for any finite coloring of $\mathbb N$, there are $\lambda,\rho\in \mathbb N$ such that there exist (infinitely many) pairs $(x,y),(u,v)\in \mathbb N^2$ such that the two sets $\{\lambda x,\lambda y,xy,\lambda(x+y)\}$ and $\{u+\rho,v+\rho,uv+\rho,u+v\}$ are monochromatic. By further employing related arguments, we can provide two different proofs for a special case of Milliken--Taylor theorem.