Homogeneous Patterns in Ramsey Theory

TL;DR

The article advances nonlinear Ramsey theory by establishing homogeneous versions of key partition-regularity phenomena. It proves that for any finite coloring of $\mathbb{Z}^+$ there exist an infinite $A$ and an arbitrarily large finite $B$ with $A\cup(A+B)\cup(A\cdot B)$ monochromatic, using ultrafilter-central-set techniques and a polynomial van der Waerden framework. It then develops a homogeneous extension of nonlinear partition regularity, showing that certain nonlinear equations like $X^2+Y^2=Z^2+P(u_1,\dots,u_n)$ are 2-regular for appropriate $P$, and provides two structural proofs (3.1 and 3.2) within this homogeneous setting. Finally, it constructs nonlinear equations with prescribed degrees of regularity, proving that for primes $p$ and integers $m,n$ with $m$ not a zero divisor in $\mathbb{Z}_n$, the equation $M_n$ is $n-1$-regular but not $n$-regular, illustrating a nonlinear analogue of Rado’s conjecture. These results deepen the understanding of when nonlinear patterns persist under colorings and offer new tools (homogeneous patterns, central sets, p-adic valuations) for arithmetic Ramsey theory.

Abstract

In this article, we investigate homogeneous versions of certain nonlinear Ramsey-theoretic results, with three significant applications. As the first application, we prove that for every finite coloring of $\mathbb{Z}^+$, there exist an infinite set $A$ and an arbitrarily large finite set $B$ such that $A \cup (A+B) \cup A \cdot B$ is monochromatic. This result resolves the finitary version of a question posed by Kra, Moreira, Richter, and Robertson regarding the partition regularity of $(A+B) \cup A \cdot B$ for infinite sets $A, B$ (see (Question 8.4, J. Amer. Math. Soc., 37 (2024))), which is closely related to a question of Erdős. As the second application, we make progress on a nonlinear extension of the partition regularity of Pythagorean triples. Specifically, we demonstrate that the equation $x^2 + y^2 = z^2 + P(u_1, \dots, u_n)$ is $2$-regular for certain appropriately chosen polynomials $P$ of any desired degree. Finally, as the third application, we establish a nonlinear variant of Rado's conjecture concerning the degree of regularity. We prove that for every $m, n \in \mathbb{Z}^+$, there exists an $m$-degree homogeneous equation that is $n$-regular but not $(n+1)$-regular. The case $m = 1$ corresponds to Rado's conjecture, originally proven by Alexeev and Tsimerman (J. Combin. Theory Ser. A, 117 (2010), and later independently by Golowich (Electron. J. Combin. 21 (2014)).

Theorems & Definitions (19)

• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Theorem 2.2
• proof : Proof of Theorem \ref{['essential']}
• proof : Proof of Theorem \ref{['task']}
• Definition 3.1
• Theorem 3.2
• proof : Proof of Theorem \ref{['pvdw']}