  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
$\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$.
In this paper, we find asymptotically tight bounds for the following two
quantitative questions.
  $\bullet$ For $n\in \mathbb{N}$, what is the longest interval $[n,f(n)]$ of
natural numbers which admits a $2$-coloring with no monochromatic solutions of
$x+y=p(z)$?
  $\bullet$ For $n\in \mathbb{N}$ 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 $g(n)$.
