Brinda Venkataramani | Search arXiv

arXiv2026

Choice principles in countable partition models

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.

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arXiv2026

Ramsey-Theoretic Finiteness in Choiceless Set Theory: The Gower's Collapse and Rainbow and Canonical Separations

We study finiteness classes arising from Ramsey-theoretic principles in set theory without the Axiom of Choice. First, we answer a question of Brot, Cao, and Fernández-Bretón concerning Gowers' $\mathbf{FIN}_k$ theorem. The $\mathbf{FIN}_k$ operation gives rise to a finiteness notion for each $k\geq 1$, but we show that this notion is independent of the value of $k$: the resulting hierarchy collapses, and every level is equivalent to $H$-finiteness, i.e., to Dedekind-finiteness of $[X]^{<ω}$. We then turn to the Rainbow Ramsey theorem and the Canonical Ramsey theorem. The corresponding failure classes $\operatorname{\mathbf{RRT-Fin}_n^m}$ and $\operatorname{\mathbf{CRT-Fin}}$ are genuine finiteness classes contained in $\operatorname{\mathbf{D-Fin}}$. We prove several inclusions placing them relative to standard Ramsey-theoretic finiteness classes of Brot--Cao--Fernández-Bretón, prove their independence from the collapsed Hindman--Gowers class, and analyze the parameter dependence of the rainbow classes through Fraenkel--Mostowski examples. The finer questions of strictness, and the full two-parameter structure of the rainbow classes, are left open where the present arguments do not settle them.

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