Partition regularity of Jp-sets in (N, +)

Prove or refute partition regularity of Jp-sets in the additive semigroup (N, +).

Background

The paper explicitly identifies partition regularity of Jp-sets as unresolved even for (N, +). A family of sets is partition regular here when every two-partition of a member has at least one cell in the same family.

Theorem 5.3 supplies only a partial result, showing that deleting a finite subset from a Jp-set in (N, +) preserves the Jp property; it does not settle partition regularity.

References

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).

— On partition and almost disjoint properties of combinatorial notions  (2501.11334 - Zhang, 20 Jan 2025) in Discussion preceding Theorem 5.3, Section 5, p. 14