Characterization of non-pathological c-coloring ideals

Determine when a c-coloring ideal is non-pathological, including the Borel or projective complexity of the collection of 2-colorings c of [N]^2 whose generated ideal ⟨hom(c)⟩ is non-pathological.

Background

The paper develops pathological and non-pathological examples of c-coloring ideals, but does not provide a general criterion. It explicitly asks both for a characterization and for the complexity of the corresponding parameter space.

References

Question 5.21. Determine when a c-coloring ideal is non-pathological. In particular, determine the the (Borel or projective) complexity of the collection {c ∈ 2[N]2 : ⟨hom(c)⟩ is non-pathological}.

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 5.21, p. 28