Partition regularity of Pythagorean pairs

TL;DR

This work resolves a central open problem in Ramsey theory by proving that every finite coloring of the positive integers yields monochromatic Pythagorean pairs $(x,y)$ with $x^2\pm y^2=z^2$ for some $z$, and extends to density-regularity statements via multiplicative-density notions. The authors develop a novel blend of Gowers uniformity for aperiodic multiplicative functions with nonlinear concentration techniques, augmented by a flexible decomposition of multiplicative functions into aperiodic and pretentious parts and by averaging over carefully chosen multiplicative Følner sequences. A key contribution is showing that level sets of finitely-valued completely multiplicative functions necessarily contain Pythagorean triples, with parametrizations supporting broader equations $ax^2+by^2=cz^2$ under square-coefficient and Rado-type constraints. The results bridge Ramsey theory, ergodic theory, and analytic number theory, offering new tools (concentration inequalities for multiplicative functions, multiplier-averaging strategies) that may apply to other nonlinear, dilation-invariant configurations and prompting further exploration of partition/density regularity in polynomial settings.

Abstract

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs, i.e., $x,y\in \mathbb{N}$ such that $x^2\pm y^2=z^2$ for some $z\in \mathbb{N}$. We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

Theorems & Definitions (87)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Remark
• Theorem 1.7
• Theorem 1.8
• Remark