Saturation of tall B-ideals by copies of c0

Determine whether every tall B-ideal admits a B-representation in a Banach space saturated by isomorphic copies of c0.

Background

Theorem 3.2 characterizes tall B-ideals through sequences with subsequences equivalent to the unit basis of c0. The unresolved strengthening asks whether the ambient space can always be chosen to be saturated by actual isomorphic copies of c0, rather than merely having sequential c0-saturation.

References

We do not know if this can be strengthened to assert that tall B-ideals can be B-represented in spaces that are saturated by isomorphic copies of c0, i.e. such that every infinite-dimensional subspace has a subspace isomorphic to c0.

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Section 1, page 3