  Inspired by Owings's problem, we investigate whether, for a given an Abelian
group $G$ and cardinal numbers $\kappa,\theta$, every colouring
$c:G\longrightarrow\theta$ yields a subset $X\subseteq G$ with $|X|=\kappa$
such that $X+X$ is monochromatic. (Owings's problem asks this for $G=\mathbb
Z$, $\theta=2$ and $\kappa=\aleph_0$; this is known to be false for the same
$G$ and $\kappa$ but $\theta=3$.) We completely settle the question for
$\kappa$ and $\theta$ both finite (by obtaining sufficient and necessary
conditions for a positive answer) and for $\kappa$ and $\theta$ both infinite
(with a negative answer). Also, in the case where $\theta$ is infinite but
$\kappa$ is finite, we obtain some sufficient conditions for a negative answer
as well as an example with a positive answer.
