Necessity of the large cardinal assumption in the minimally unbounded case

Ascertain whether the existence of the first ω‑Erdős cardinal κ(ω) is necessary for proving that T(P,≤,δ) is not Borel complete when (P,≤,δ) is minimally unbounded. Equivalently, determine whether the non‑Borel completeness of T(P,≤,δ) for all minimally unbounded triples (P,≤,δ) can be established in ZFC, or whether the large cardinal assumption cannot be eliminated.

Background

The paper studies the Borel complexity of theories T(P,≤,δ) arising from countable down‑finite posets equipped with finite splitting parameters δ. A central classification divides cases into bounded, unbounded, and minimally unbounded, with minimally unbounded triples forming a delicate boundary. In Theorem 1.3 the authors use the existence of the first ω‑Erdős cardinal κ(ω) to show that in the minimally unbounded case, T(P,≤,δ) is not Borel complete.

This proof leverages results on Schroeder–Bernstein properties that depend on κ(ω). The authors highlight that it remains unknown whether such a large cardinal assumption is essential, raising the question of whether the non‑Borel completeness result can be achieved in ZFC or if the assumption is inherently required.