Large Cardinal Foundations of MAX Universal Baireness

Definitions and mathematical statement

Write $\mathrm{UB}$ for the universally Baire sets referred to in the source record. For terminology, a subset $A$ of a topological space $X$ is universally Baire if, for every compact Hausdorff space $K$ and every continuous map $f:K\to X$, the set $f^{-1}[A]$ has the Baire property in $K$. A set has the Baire property if its symmetric difference with some open set is meagre; a meagre set is a countable union of nowhere dense sets. The record does not specify the ambient spaces, coding conventions, or interpretation of $\mathrm{UB}$ in forcing extensions required for its intended axiom.

The source describes $\operatorname{MAX}(\mathrm{UB})$ as asserting a proper class of Woodin cardinals together with universal-Baireness and generic-elementarity conditions. Its explicitly identified large-cardinal component is \[ \forall\alpha\in\operatorname{Ord}\;\exists\delta\, \left(\alpha<\delta\ \land\ \delta\text{ is a Woodin cardinal}\right), \] where $\operatorname{Ord}$ denotes the class of ordinals. The additional conditions are referred to as "stated" in the source statement but are not actually supplied in this record. Consequently, this description does not give a complete definition of $\operatorname{MAX}(\mathrm{UB})$.

Source question, with unresolved specifications. Is $\operatorname{MAX}(\mathrm{UB})$ a direct consequence of suitable large-cardinal axioms?

The intended question concerns implication from large-cardinal axioms themselves, as distinguished in the supplied assessment from obtaining the axiom in a forcing extension. However, the record specifies neither the candidate large-cardinal hypotheses nor the background theory or precise meaning of "direct consequence." Together with the omitted conditions defining $\operatorname{MAX}(\mathrm{UB})$, these omissions prevent a fully specified formal implication. This draft preserves the question and its explicit proper-class requirement without selecting additional hypotheses or supplying the missing axiom.

Short English statement

The question asks whether suitable large-cardinal axioms directly imply $\operatorname{MAX}(\mathrm{UB})$, which the source says includes a proper class of Woodin cardinals. The defining universal-Baireness and generic-elementarity clauses, and the intended sufficient hypotheses, are missing from the supplied record. The supplied status is open; this is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30004450$, title "Large Cardinal Foundations of MAX Universal Baireness," collection "Oberwolfach Reports Open Problems," category "Set Theory," exact source problem code catalogue entry. The supplied proposed year is 2020; no individual proposer is identified.
• Supplied report citation: Set Theory (2021), Oberwolfach Reports, pp. 797--855, DOI 10.4171/owr/2020/14. https://doi.org/10.4171/owr/2020/14. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source: Viale (2021), https://arxiv.org/abs/2101.07573. No paper title or specific definition, theorem, or page reference is supplied.
• Supplied literature source: "Oberwolfach Report 14/2020," https://doi.org/10.4171/owr/2020/14.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that $\operatorname{MAX}(\mathrm{UB})$ is known in generic extensions obtained by collapsing a supercompact and is preserved by further set forcing, while implication from suitable large-cardinal axioms alone remains open. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The record expressly requests independent verification of time-sensitive status and cited claims; its summary does not specify the collapse, auxiliary hypotheses, or the precise scope of the preservation assertion.
• The supplied statement status is corrected_verified; its accompanying explanation limits the changes to formula delimiters, notation, grammar, and typography, with mathematical content and scope preserved. This label does not establish mathematical correctness or independent source verification for this draft.