A central theme of arithmetic Ramsey theory is that every finite coloring of the natural numbers should contain monochromatic solutions to certain algebraic equations. While Rado's theorem gives a complete understanding in the linear setting, the nonlinear case remains largely mysterious. A prominent example is the partition regularity of Pythagorean triples, a question posed by Erdos and Graham in the 1970s. In joint work with Moreira and Klurman, building on earlier work with Host, we established partition regularity for Pythagorean pairs, where two of the three variables are required to have the same color, as well as for several other quadratic patterns. The key ingredients are multiplicative Fourier analysis and structure-randomness decompositions for multiplicative functions. For full Pythagorean triples, however, this approach breaks down. This leads to a new ergodic-theoretic framework based on multiplicative actions, which reduces the problem to a multiple recurrence statement. I will describe these developments, the new ergodic ideas that enter, and the challenges that remain.