Partition regularity of Pythagorean pairs

TL;DR

This paper proves that any finite coloring of positive integers necessarily contains monochromatic Pythagorean pairs, and that certain partitions derived from multiplicative functions also contain Pythagorean triples, advancing understanding in Ramsey theory.

Contribution

It establishes the partition regularity of Pythagorean pairs and introduces a novel approach combining Gowers uniformity with concentration estimates of multiplicative functions.

Findings

Every finite coloring of positive integers contains monochromatic Pythagorean pairs.

Partitions from level sets of multiplicative functions contain Pythagorean triples.

The proof combines Gowers uniformity with concentration estimates of multiplicative functions.

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.