Markov's equation is not partition regular

Markov’s equation x^2 + y^2 + z^2 = 3xyz is a widely studied topic in number theory, and the structure of its solutions has profound connections with mathematical fields such as combinatorics, hyperbolic geometry, approximation theory, and cluster algebras. In this paper, we prove that Markov’s equation is not partition regular, which also confirms a necessary condition for the Uniqueness Conjecture.