B-representation in spaces saturated by copies of c0

Establish whether every tall B-ideal can be B-represented in a Banach space saturated by isomorphic copies of c0.

Background

Theorem 3.2 characterizes tall B-ideals through sequences whose subsequences contain further subsequences equivalent to the unit basis of c0. The stronger property that the ambient space itself is saturated by isomorphic copies of c0 is not established.

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 Introduction, p. 3