Monochromatic polynomial sumset structures on $\mathbb{N}$

TL;DR

The paper investigates Ramsey-type questions for monochromatic polynomial sumset structures on $\mathbb{N}$ and proves that for any $r\in\mathbb{N}$, distinct $a,b$, and any 2-coloring of $\mathbb{N}$, there exist $B$ with $|B|=r$ and $C$ with $|C|=\infty$ such that a single color contains both $B+aC$ and $B+bC$. It leverages a blend of ergodic theory (polynomial multiple recurrence, Kronecker factors), combinatorial methods, and dynamical systems (shift spaces, joinings) to obtain the positive result, while providing explicit counterexamples to show the limits of such Ramsey phenomena in other configurations. The work clarifies which polynomial-sum configurations persist under 2-colorings and which fail, advancing the understanding of polynomial sumsets in Ramsey theory and their density-entropy connections. The results have implications for the structure of monochromatic polynomial progressions in additive combinatorics and highlight the role of dynamical-structural tools in Ramsey-type problems on the integers.

Abstract

In the paper, we search for monochromatic infinite additive structures involving polynomials over $\mathbb{N}$. It is proved that for any $r\in \mathbb{N}$, any two distinct natural numbers $a,b$, and any $2$-coloring of $\mathbb{N}$, there exist two sets $B,C\subset \mathbb{N}$ with $|B|=r$ and $|C|=\infty$ such that there exists some color containing $B+aC$ and $B+bC$.

Theorems & Definitions (31)

• Theorem 1.1
• Proposition 1.2
• Remark 1.4
• Proposition 1.5
• Proposition 1.6
• Theorem 1.7
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Corollary 2.4