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

TL;DR

This paper proves the existence of monochromatic configurations involving sums and products with additive or multiplicative shifts in any finite coloring of natural numbers, and provides new proofs for a related combinatorial theorem.

Contribution

It introduces novel results on monochromatic sum-product configurations with shifts and offers two different proofs of a special case of the Milliken--Taylor theorem.

Findings

Existence of monochromatic sets with sum and product structures under finite colorings

Construction of specific lambda, rho in N for infinitely many pairs satisfying the conditions

Two distinct proofs of a particular case of the Milliken--Taylor theorem

Abstract

In this paper we prove that for any finite coloring of N there are lambda,rho in N such that infinitely many pairs (x,y),(u,v) in N^2 satisfy the sets {lambda x, lambda y, x y, lambda(x+y)} and {u+rho, v+rho, u v+rho, u+v} being monochromatic. Using related arguments we also give two different proofs of a special case of the Milliken--Taylor theorem.