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