Is x^(2)+y^(2)=z^(2) partition regular

Definitions and mathematical statement

A finite coloring of the positive integers is a map $c:\mathbb Z_{>0}\to\{1,\ldots,r\}$, where $r\ge1$ is finite but otherwise arbitrary. A monochromatic Pythagorean triple is a triple of positive integers $x,y,z$ satisfying \[ x^2+y^2=z^2, \qquad c(x)=c(y)=c(z). \] Is the Pythagorean equation partition regular over the positive integers, meaning that every such finite coloring has a monochromatic solution?

The three integers are necessarily distinct: positivity gives $z>x,y$, and $x=y$ would make $z/x=\sqrt2$ rational. They need not form a primitive triple; imposing $\gcd(x,y,z)=1$ would be an additional restriction. The number of colors is fixed within each coloring, but the assertion must hold for every finite $r$.

The two-color case has already been established and is not the remaining general question. Likewise, conclusions making only two of the three entries monochromatic do not establish full partition regularity. Equivalently, for each $r$ one asks for a finite $N(r)$ such that every $r$-coloring of $[N(r)]$ contains the required triple; the finite and infinite formulations are related by the usual compactness argument for finite colorings.

Short English statement

Does every finite coloring of the positive integers contain a monochromatic Pythagorean triple?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100061 (AMR-030-0061), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Combinatorial Number Theory, source-order bullet 61. Original source indexed by AMR-030-0061. https://people.math.sc.edu/cooper/combprob.html.
• S3 Heule, Kullmann and Marek, Solving and Verifying the Boolean Pythagorean Triples problem. https://arxiv.org/abs/1605.00723.