Optimal bounds on the polynomial Schur's theorem

Liu, Pach and S\'andor recently characterized all polynomials p (z) such that the equation x+y=p (z) is 2-Ramsey, that is, any 2-coloring of N contains infinitely many monochromatic solutions for x+y=p (z). In this paper, we find asymptotically tight bounds for the following two quantitative questions. For n N, what is the longest interval n, f (n) of natural numbers which admits a 2-coloring with no monochromatic solutions of x+y=p (z)? For n N and a 2-coloring of the first n integers n, what is the smallest possible number g (n) of monochromatic solutions of x+y=p (z)? Our theorems determine f (n) up to a multiplicative constant 2+o (1), and determine the asymptotics for g (n).