Erdős–Graham conjecture on monochromatic Pythagorean triples

A finite colouring of $\mathbb{N}$ is a colouring with finitely many colours. A Pythagorean triple is a triple $(x,y,z)\in\mathbb{N}^3$ satisfying $x^2+y^2=z^2$, and it is monochromatic when all three entries have the same colour.

Erdős–Graham conjecture. Any finite colouring of $\mathbb{N}$ has a monochromatic Pythagorean triple.

The necessary direction is not at issue: the conjecture asserts the existence of such a triple for every finite colouring. The paper describes this problem as remaining unsolved and establishes a special case using completely multiplicative functions.