  We study the Ramsey properties of equations $a_1P(x_1) + \cdots + a_sP(x_s) =
b$, where $a_1,\ldots,a_s,b$ are integers, and $P$ is an integer polynomial of
degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we show that
Rado's criterion and an intersectivity condition completely characterise which
equations of this form admit monochromatic solutions with respect to an
arbitrary finite colouring of the positive integers. Furthermore, we obtain a
Roth-type theorem for these equations, showing that they admit non-constant
solutions over any set of integers with positive upper density if and only if
$b= a_1 + \cdots + a_s = 0$. In addition, we establish sharp asymptotic lower
bounds for the number of monochromatic/dense solutions (supersaturation).
