(ω1, ℵω1)-compactness of ℵω1

Determine whether the cardinal ℵω1 can be (ω1, ℵω1)-compact.

Background

The paper introduces (δ, λ)-compactness as a hierarchy of infinitary logical compactness properties and characterizes these properties using compactness for list colourings. The authors specifically identify unresolved questions about (ω1, κ)-compact cardinals below the general results established in the paper.

The first question asks whether the particular cardinal ℵω1 can satisfy its corresponding (ω1, κ)-compactness property, addressing the possible existence of such compactness at this explicitly specified cardinal.

References

The most obvious questions that naturally arise from the results of our paper seem to be those concerning the properties of our hierarchy of compact cardinals, and we ask what we think might be very interesting sample questions regarding (ω1, κ)-compact cardinals κ below: Question 29. * Can κ = ℵω1 be (ω1, κ)-compact?

— Large cardinal characterizations via compactness for list colourings  (2608.20883 - Feller et al., 21 Aug 2026) in Question 29, Open Questions, Appendix A, p. 18