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