Partition regularity of the Pythagorean equation

Determine whether the Pythagorean equation x² + y² = z² is partition regular over the natural numbers.

Background

Partition regularity asks whether every finite colouring of the natural numbers contains a monochromatic solution to a specified equation. The paper notes that the linear case is well understood, whereas the corresponding polynomial theory remains unsettled. The Pythagorean equation is presented as a longstanding unresolved example, although the paper’s main focus is the stronger notion of Ramsey partition regularity.

References

For instance, whether the Pythagorean equation x2 + y2 = z2 is partition regular is still unknown, and one of the long-standing open problems in the area.

— Ramsey's witnesses  (2503.09246 - Nasso et al., 12 Mar 2025) in Section 1, Introduction, p. 2

In particular, the following seemingly simple question posed by Erdős and Graham remains open: Question 1.3. [15, 16] Is the equation x² + y² = z² partition regular?

— Homogeneous Patterns in Ramsey Theory  (2501.17203 - Adhikari et al., 28 Jan 2025) in Question 1.3 and surrounding discussion, Section 1, p. 2

For instance, whether the Pythagorean equation x2 + y2 = z2 is partition regular is still unknown, and one of the long-standing open problems in the area.

— Ramsey's witnesses  (2503.09246 - Nasso et al., 12 Mar 2025) in Section 1, Introduction, p. 2