Partition regularity of Pythagorean triples

Determine whether the equation x²+y²=z² is partition regular over the positive integers; that is, establish whether every finite coloring of the positive integers contains a monochromatic positive-integer solution.

Background

The paper presents the partition regularity of the Pythagorean equation as a central unresolved problem in nonlinear Ramsey theory. It notes that computational work has confirmed the result for two-colorings, while the question for arbitrary finite colorings remains open.

References

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, Section 1, Introduction, p. 2