Polynomial Schur’s Theorem

We resolve the Ramsey problem for {x,y,z: x + y = p(z)} for all polynomials p over ℤ. In particular, we characterise all polynomials that are 2-Ramsey, that is, those p(z) such that any 2-colouring of ℕ contains infinitely many monochromatic solutions for x+y=p(z). For polynomials that are not 2-Ramsey, we characterise all 2-colourings of ℕ that are not 2-Ramsey, revealing that certain divisibility barrier is the only obstruction to 2-Ramseyness for x + y = p(z)