Owings-like results for infinitely many colours or finite monochromatic sets

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers κ,θ, every colouring c:Gθ yields a subset X⊆ G with |X|=κ such that $X+X$ is monochromatic. (Owings's problem asks this for G= Z, θ=2 and κ=₀; while Hindman proved it false for the same G and κ but θ=3.) We completely settle the question (in the affirmative whenever G is infinite) for finite κ and θ, and also (in the negative) for κ and θ both infinite. We also show that, in the case where θ is infinite but κ is finite, the answer depends both on the cardinality of G and in its algebraic structure.