Spines in vacillating FAC posets by cardinality

Classify the infinite cardinals κ for which every vacillating poset P with no infinite antichain and |P| = κ admits a spine, i.e., a chain C and a partition of P into antichains such that C meets every antichain of the partition.

Background

The main theorem of the paper proves that countable vacillating FAC posets admit spines. The authors note that their arguments are sensitive to cardinality and ask whether similar results hold at larger cardinalities.

A positive characterization by cardinality would extend the main result beyond the countable setting and clarify the role of vacillation in infinite contexts.

References

The proofs presented here were in places highly sensitive to the cardinality of $P$, and so we ask whether larger cardinalities present a genuine issue or not. Let $P$ be a vacillating FAC poset of cardinality $\kappa$. For which $\kappa$ must $P$ have a spine?

— A resolution of the Aharoni-Korman conjecture (2411.16844 - Hollom, 25 Nov 2024) in Question 7.2, Section 7 (Concluding remarks and open problems)