Existence of hereditarily pathological Fσ ideals

Construct an Fσ ideal such that every restriction to a positive set is pathological.

Background

The paper introduces hereditary non-pathology through restrictions to positive sets and proves local non-pathology for certain Q-coloring ideals. The dual existence question—whether an Fσ ideal can be pathological on every positive restriction—remains open.

References

Question 5.33. Does there exist a hereditarily pathological Fσ-ideal? Specifically, is there an Fσ-ideal such that every restriction to a positive set is pathological?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 5.33, p. 31