Composite Ramsey theorems via trees

TL;DR

This paper proves that certain Ramsey properties are preserved under composition, and demonstrates the existence of large monochromatic sets with combined additive and multiplicative structures in finite colorings of natural numbers.

Contribution

It introduces a novel tree-based color focusing technique and generalizes existing Ramsey theorems to broader algebraic structures.

Findings

Compositions of specific Ramsey families remain Ramsey.

Existence of large monochromatic sets with combined sum and product structures.

Generalization of Bergelson and Moreira's Ramsey theorem beyond fields.

Abstract

We prove a theorem ensuring that the compositions of certain Ramsey families are still Ramsey. As an application, we show that in any finite coloring of $\mathbb{N}$ there is an infinite set $A$ and an as large as desired finite set $B$ with $(A+B)\cup (AB)$ monochromatic, answering a question from a recent paper of Kra, Moreira, Richter, and Robertson. In fact, we prove an iterated version of this result that also generalizes a Ramsey theorem of Bergelson and Moreira that was previously only known to hold for fields. Our main new technique is an extension of the color focusing method that involves trees rather than sequences.