Partition regularity of PP-rich sets in uncountable semigroups

Establish whether PP-rich sets have partition regularity in uncountable semigroups.

Background

The paper recalls that PP-rich sets are partition regular in (N, +): every two-color partition of a PP-rich set has a PP-rich cell. It then proves partition and almost disjoint results for PP-rich sets in (N, +), but does not determine whether partition regularity persists when the underlying semigroup is uncountable.

References

When the semigroup is uncountable, we do not know whether PP-rich sets still have partition regularity. So we close this section with following questions. Question 5.8. Do PP-rich sets have partition regularity in uncountable semigroups?

— On partition and almost disjoint properties of combinatorial notions  (2501.11334 - Zhang, 20 Jan 2025) in Question 5.8, Section 5, p. 15