Pythagorean Partition-Regularity and Ordered Triple Systems with the Sum Property

TL;DR

This paper investigates the coloring of natural numbers to avoid monochromatic Pythagorean triples, showing that certain ordered triple systems with a sum property cannot contain Steiner triple systems, advancing understanding of Pythagorean partition-regularity.

Contribution

It proves that ordered triple systems with a sum property do not contain Steiner triple systems, providing new insights into the structure of Pythagorean triples and related combinatorial systems.

Findings

Pythagorean triples do not contain the Fano plane.

Ordered triple systems with the sum property exclude Steiner triple systems.

Supports the conjecture about coloring naturals to avoid monochromatic Pythagorean triples.

Abstract

Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is even open for two colors. A natural strategy is to show that some small nonbipartite triple systems cannot be realized as a family of Pythagorean triples. It suffices to consider partial triple systems (PTS's), and it is therefore natural to consider the Fano plane, the smallest nonbipartite PTS. We show that the Pythagorean triples do not contain any Fano plane. In fact, our main result is that a much larger family of "ordered" triple systems (viz. those with a certain "sum property") do not contain any Steiner triple system (STS).