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

Abstract

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 Fern\'andez-Bret\'on 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 $[X]^{<\omega}$. 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--Fern\'andez-Bret\'on, 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.