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 $g(n)$.