Characterization of groups with finite monochromatic sumsets under infinitely many colours (Question 9)

Characterize, in terms of cardinal invariants such as |G|, |G2|, |G4|, and/or |G/Gd| with Gd = {x ∈ G : d x = 0}, the Abelian groups G for which (either for all finite n, or for some specified n) the property G → (n)θ+· holds.

Background

Proposition 7 provides large examples (direct sums of Z/4Z of sufficiently large cardinality) where, for each fixed finite n, one can find a set X of size 2n with X + X monochromatic under any θ-colouring. Combined with Theorem 6, which gives broad negative cases, this suggests a structural classification problem.

The authors suggest that invariants related to 2- and 4-torsion (and quotients by d-torsion) may control the property, and they explicitly pose Question 9 to seek a precise characterization.

References

Obtaining more precise information along this lines seems to be an interesting question that we leave open. Question 9. Is it possible to characterize (in terms of |G|, |G |, |G | and/o2 possi4ly |G/G |, where G = {x ∈ G dx = 0}) precisely those Abelian groups G satisfying (whether for all n, or for d ·+* some specific one) G → (n) θ ?

— Owings-like theorems for infinitely many colours or finite monochromatic sets (2402.13124 - Fernández-Bretón et al., 20 Feb 2024) in Section 4 (Infinitely many colours, finite monochromatic sets), Question 9