Partition and almost disjoint properties of Jp-sets

Determine whether Jp-sets possess any partition or almost disjoint properties, including partition regularity.

Background

Jp-sets are the polynomial analogue of J-sets and are central to the polynomial extension of the central sets theorem. The paper extends the definition of Jp-sets to commutative cancellative semigroups and proves a partial result: removing a finite subset from a Jp-set in (N, +) leaves a Jp-set.

However, the authors obtain no corresponding infinite partition or almost disjoint theorem for Jp-sets, even in (N, +). They also note that partition regularity of Jp-sets is an open question in the cited work, making the broader collection of partition and almost disjoint properties unresolved.

References

After getting the partition and almost disjoint properties of Cp-sets, it is natural to consider Jp-sets. Unfortunately, it is difficult to obtain analogous results for Jp-sets. And the partition regularity of Jp-sets is also hard to obtain (which is an open question [2, Question 17] for the case S = N). But we still have the following result, a partial answer of [2, Question 17].

— On partition and almost disjoint properties of combinatorial notions  (2501.11334 - Zhang, 20 Jan 2025) in Question 5.10, Section 5, p. 15

Question 5.10. Do Jp-sets have any partition or almost disjoint properties? Or partition regularity?

— On partition and almost disjoint properties of combinatorial notions  (2501.11334 - Zhang, 20 Jan 2025) in Question 5.10, Section 5