Eliminating Special Aronszajn Trees Along Radin Clubs

Definitions and mathematical statement

Let $\lambda$ be an infinite regular cardinal. A $\lambda$-tree is a partially ordered set $(T,<_{T})$ such that the predecessors of each node are well ordered, the tree has height $\lambda$, and each level $T_\alpha$, for $\alpha<\lambda$, has cardinality less than $\lambda$. A cofinal branch is a linearly ordered subset whose node heights are unbounded in $\lambda$. A $\lambda$-Aronszajn tree is a $\lambda$-tree with no cofinal branch. The tree property at $\lambda$, denoted here by $\operatorname{TP}(\lambda)$, is the assertion that no $\lambda$-Aronszajn tree exists.

For an infinite cardinal $\mu$, a $\mu^{+}$-Aronszajn tree $T$ is special if there is a function $f:T\to\mu$ such that \[ s<_{T}t\implies f(s)\ne f(t). \] Equivalently, $T$ is the union of at most $\mu$ antichains. Write $\operatorname{NSAT}(\mu^{+})$ for the assertion that no special $\mu^{+}$-Aronszajn tree exists. Thus $\operatorname{TP}(\mu^{+})$ implies $\operatorname{NSAT}(\mu^{+})$, whereas the two assertions express different targets.

For a limit ordinal $\kappa$, a subset $C\subseteq\kappa$ is closed unbounded if it is unbounded in $\kappa$ and, for every limit ordinal $\delta<\kappa$, \[ \sup(C\cap\delta)=\delta\implies\delta\in C. \]

The supplied record presents two formulations that must be retained separately. Its general question asks whether a cited consistency result excluding special Aronszajn trees at a prescribed collection of regular cardinals can be strengthened to give the tree property at every cardinal in that collection. The collection, the original consistency theorem, and its hypotheses are not supplied, so this does not specify a complete relative consistency statement.

Its explicit Radin-forcing question asks whether there is a Radin forcing $\mathbb P$ adding a club $C\subseteq\kappa$ such that every cardinal $\mu\in C$ has no special $\mu^{+}$-Aronszajn tree. Under the conventional interpretation that the cardinals under discussion are infinite, the requested extension satisfies \[ V[G]\models \text{$C\subseteq\kappa$ is closed unbounded} \quad\text{and}\quad \forall\mu\in C\, \left(\mu\text{ is an infinite cardinal} \implies\operatorname{NSAT}(\mu^{+})\right), \] where $G$ is $\mathbb P$-generic and cardinalhood and successors are computed in $V[G]$. The source says "every cardinal" without explicitly imposing infinitude; its treatment of finite cardinals is unspecified.

A corresponding club-wide formulation of the stronger target would replace $\operatorname{NSAT}(\mu^{+})$ by $\operatorname{TP}(\mu^{+})$. The explicit source question itself asks only for the absence of special trees. The record does not determine whether these are successive questions or competing descriptions of one intended target. It also does not specify the hypotheses on $\kappa$, the permitted Radin forcing, or the consistency strength. These omissions remain unresolved.

Short English statement

The record asks whether a consistency result excluding special Aronszajn trees can be strengthened to exclude all Aronszajn trees. Its concrete Radin-forcing formulation instead asks for a club $C\subseteq\kappa$ with no special $\mu^{+}$-Aronszajn tree at each cardinal $\mu\in C$. These targets differ, and the underlying consistency assumptions are missing. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003410$, exact source problem code catalogue entry, title "Eliminating Special Aronszajn Trees Along Radin Clubs," Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2017.
• Supplied source citation: Set Theory (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/11, pp. 527--589. https://doi.org/10.4171/owr/2017/11. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's 2018 title and its 2017 DOI/proposed-year information are retained without reconciliation.
• Supplied literature source: "OWR Set Theory report." https://ems.press/content/serial-article-files/46674.
• Supplied literature source: "Diagonal supercompact Radin forcing." https://doi.org/10.1016/j.apal.2020.102828.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that diagonal supercompact Radin forcing gives clubs with failure of the Singular Cardinal Hypothesis (SCH) and weak square, while the stronger club-wide absence of special successor Aronszajn trees asked here is not established. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly and that its cleaned statement preserves the target while restoring definitions and removing source-dependent wording. These are supplied verification claims; no source visit or independent verification is asserted in this draft.