Choice principles in countable partition models

@misc{venkataramani2026choice,
  title = {{Choice principles in countable partition models}},
  author = {Brinda Venkataramani},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29752},
  eprint = {2610.29752},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29752},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Choice principles in countable partition models
AU  - Brinda Venkataramani
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29752
UR  - https://arxiv.org/abs/2610.29752
AB  - We study the countable partition model over a set of atoms of arbitrary uncountable cardinality κ. Its supports are partitions of the atoms into countably many blocks, and a permutation fixes a support when it preserves each block setwise. This model was proposed by Bruce and studied by Banerjee. We prove that dependent choice holds for every κ, although the filter of subgroups is not closed under countable intersections. Choice for well-orderable families holds exactly when κ<ℵ_ω_1; when κ≥ℵ_ω_1, an ω_1-indexed family witnesses its failure. The Hartogs and Lindenbaum numbers of the set of atoms are both ω_1. The proofs use permutations matching blocks of equal cardinality and establish shift completeness of the filter. Together with Banerjee's failure of DC_ω_1, these results determine which dependent choice principles hold in the model.
ER  -