Canonical equivalence relations on $\mathrm{FIN}^{[\infty]}_2$

Abstract

Answering a question of Todorcevic, we prove higher-dimensional canonization theorems for the topological Ramsey space $\mathrm{FIN}^{[\infty]}_2$. Our results build upon the work of Lopez-Abad and continue the line of research initiated by Erd\H{o}s and Rado, and further developed by Pudl\'ak and R\"odl, Pr\"omel and Voigt, Taylor, and Klein and Spinas. We identify the canonical functions on fronts of $\mathrm{FIN}^{[\infty]}_2$ and develop an extension of the separating-mixing technique of Pr\"omel and Voigt to canonize arbitrary functions $g:\mathcal{F}\to\omega$, where $\mathcal{F}$ is a front of $\mathrm{FIN}^{[\infty]}_2$. We further extend our canonization theorem to arbitrary Borel maps $g:\mathrm{FIN}^{[\infty]}_2\to\mathbb{R}$, establishing new canonical Ramsey theorems for Polish spaces. In particular, we provide a complete classification of the canonical functions on $\mathrm{FIN}^{[n]}_2$, together with an explicit formula for their number as a function of $n$.