Partition, almost disjoint, and partition-regularity properties of Jp-sets

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

Background

The paper introduces Jp-sets as polynomial analogues of J-sets and proves a partial result in (N, +): removing a finite subset from a Jp-set leaves a Jp-set. However, it obtains no corresponding partition or almost disjoint theorem, even for (N, +).

The authors also note that partition regularity of Jp-sets is an open question for (N, +), previously identified as such in the cited literature. The final question asks simultaneously whether any partition or almost disjoint phenomena hold and whether the family of Jp-sets is partition regular.