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 Fern'andez-Bret'on concerning Gowers’ ${\mathbf{F}\mathbf{I}\mathbf{N}}_k$ theorem. The ${\mathbf{F}\mathbf{I}\mathbf{N}}_k$ operation gives rise to a finiteness notion for each $k\ge 1$, 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 $\mathrm{}$ and $\mathrm{}$ are genuine finiteness classes contained in $\mathrm{}$. 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.