Choice principles in countable partition models

Abstract

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 $κ<\aleph_{ω_1}$; when $κ\ge\aleph_{ω_1}$, 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 $\mathsf{DC}_{ω_1}$, these results determine which dependent choice principles hold in the model.