Optimal bounds on the polynomial Schur's theorem

Liu, Pach and Sándor 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).