Absence of 4-torsion and failure of 2-element monochromatic sumsets with countably many colours

Establish whether every uncountable Abelian group G with no elements of order 4 satisfies G ⟶̸ (2)ℵ0+·; that is, determine whether there always exists a countably infinite colouring c: G → ℵ0 such that for every distinct x,y ∈ G, the set {2x, 2y, x + y} is not monochromatic.

Background

Theorem 6 shows that G ⟶̸ (2)ℵ0+* holds under several conditions (e.g., G countable; G torsion with no elements of order 4; or G with no elements of order 2). The authors ask whether the weaker hypothesis—no elements of order 4—already suffices for all uncountable groups.

They present examples illustrating subtleties related to embeddings, and then pose Question 8 to isolate the core uncertainty. A note added in print reports that this question has since been answered affirmatively by Leader and Williams (Leader et al., 4 Jul 2024).