Jónsson Cardinals, Trace Filters, and Indecomposable Ultrafilters

Definitions and mathematical statement

For a set $X$, write $[X]^{<\theta}$ for its subsets of cardinality less than $\theta$ and $[X]^\theta$ for those of cardinality $\theta$. A cardinal $\kappa$ is regular if $\operatorname{cf}\kappa=\kappa$, and is Jónsson if every $f:[\kappa]^{<\omega}\to\kappa$ has a set $A\in[\kappa]^\kappa$ with $f``[A]^{<\omega}\ne\kappa$. A proper filter $\mathcal F$ on $\kappa$ is uniform if every member has cardinality $\kappa$. Let $\kappa$ be a regular Jónsson cardinal. The first two source questions are:

1. Do there exist $c:[\kappa]^{<\kappa}\to\kappa$ and a uniform filter $\mathcal F$ on $\kappa$ such that \[\forall A\in[\kappa]^\kappa:\qquad c``[A]^{<\omega}\in\mathcal F?\] The domain is deliberately $[\kappa]^{<\kappa}$, whereas the image condition uses only finite subsets of $A$, exactly as in the source.
2. Can a $C$-system on $\kappa$ have a nontrivial trace filter? The extract defines neither construction. A usual $C$-sequence assigns a club $C_\delta\subseteq\delta$ at limit ordinals, but this does not identify the source's precise $C$-system, its trace operation, or the intended nontriviality condition.

The third question has separately named cardinals: let $\mu$ be singular and strong limit, meaning $\operatorname{cf}\mu<\mu$ and $2^\nu<\mu$ for every cardinal $\nu<\mu$, and put $\lambda=\mu^+$. Is existence of the source's indecomposable ultrafilter on $\lambda$ compatible with \[2^\mu>\mu^+?\] Here compatibility asks for simultaneous consistency, rather than a cardinal-arithmetic implication proved from the stated hypotheses. Indecomposability needs a parameter or range not supplied in the extract. For example, for an infinite cardinal $\theta$, an ultrafilter $\mathcal U$ on $\lambda$ is $\theta$-indecomposable if every $f:\lambda\to\theta$ admits $X\subseteq\theta$ of size less than $\theta$ with $f^{-1}(X)\in\mathcal U$; equivalently no such map pushes $\mathcal U$ to a uniform ultrafilter on $\theta$. The source does not say which $\theta$ or range is intended, or whether the ultrafilter must be uniform/nonprincipal. Those restrictions cannot be omitted without changing the intended question, since principal ultrafilters satisfy the displayed indecomposability condition automatically. No identification $\kappa=\lambda$ is stated. The supplied 2026-08-21 triage labels the bundle open, describes square-bracket partition/filter questions at successors of singulars, and reports no complete resolution located.

Short English statement

The bundle asks for a coloring whose finite-subset images belong to one uniform filter, a nontrivial trace filter from a $C$-system, and consistency of an indecomposable ultrafilter at a singular successor with failure of the singular cardinal hypothesis there. The trace construction and indecomposability range are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000905, source problem catalogue entry, Jónsson Cardinals, Trace Filters, and Indecomposable Ultrafilters, proposed year 2008; no individual proposer metadata is supplied.
• Set Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/02, pp. 79--118: https://doi.org/10.4171/owr/2008/02. Supplied literature label and attribution: Eisworth, open problems in combinatorial set theory.