Large Cardinals Supporting Brown–Adams Pairs

Definitions and mathematical statement

Let $\mathcal T$ be the triangulated category intended by the report. The supplied literature assessment places the question in the class of well-generated triangulated categories, but the extracted statement does not give the hypotheses on $\mathcal T$ or define a Brown--Adams pair.

Write $\mathcal A(\mathcal T)$ for the class of cardinals $\alpha$ satisfying the report's Brown--Adams-pair condition for $(\mathcal T,\alpha)$. This notation records the missing predicate rather than defining it. The two requested tasks are:

1. Determine whether $\mathcal A(\mathcal T)$ is unbounded in the cardinals, namely whether \[ \forall\kappa\text{ a cardinal}\ \exists\alpha\text{ a cardinal}: \quad\alpha>\kappa\quad\text{and}\quad (\mathcal T,\alpha)\text{ is a Brown--Adams pair}. \] If the assessment's universal scope is intended, this is asked for every well-generated $\mathcal T$, with $\alpha$ allowed to depend on $\mathcal T$ and $\kappa$.
2. Characterize the permissible cardinals: for a given admissible $\mathcal T$, determine precisely which $\alpha$ belong to $\mathcal A(\mathcal T)$.

Unboundedness is weaker than saying that every sufficiently large cardinal is permissible, and neither condition is substituted for the other. Likewise, "arbitrarily large" describes size in the cardinal ordering and does not itself impose a large-cardinal axiom such as inaccessibility.

A complete formulation requires the report's definition of Brown--Adams pair, including its representability requirements, test objects or functor categories, any distinction between representing objects and morphisms, and the regularity or size requirements on $\alpha$. The extraction supplies none of these. No particular transfinite Adams-representability convention or $\alpha$-compactness definition is silently used to replace the missing predicate.

Short English statement

The question asks whether the cardinals making a triangulated category into a Brown--Adams pair are unbounded, and asks exactly which cardinals work. The pair condition, category hypotheses, and permitted cardinal class must be recovered before either task is mathematically definite.

Sources

• Local UnsolvedMath registry, permanent ID 30000428, code catalogue entry, "Large Cardinals Supporting Brown--Adams Pairs." Proposed year 2006; proposer unspecified.
• Mini-Workshop: Thick Subcategories --- Classifications and Applications (2006), Oberwolfach Reports, pp. 461--510, DOI 10.4171/owr/2006/08; https://doi.org/10.4171/owr/2006/08.
• Supplied reference "Muro--Raventos transfinite representability": https://doi.org/10.1016/j.aim.2016.01.009.
• Statement status corrected_verified; the supplied note says cardinal and triangulated-category notation was restored, preserving both existence and classification questions.
• Literature triage checked 2026-08-21: open. It reports criteria and examples from transfinite Adams representability but says the arbitrarily-large-cardinal question for every well-generated triangulated category remains open. Exact conventions and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.