Canonical equivalence relations on ${\mathrm{F}\mathrm{I}\mathrm{N}}_2^{[\mathrm{\infty }]}$

Answering a question of Todorcevic, we prove higher-dimensional canonization theorems for the topological Ramsey space ${\mathrm{F}\mathrm{I}\mathrm{N}}_2^{[\mathrm{\infty }]}$. 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 ${\mathrm{F}\mathrm{I}\mathrm{N}}_2^{[\mathrm{\infty }]}$ 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 ${\mathrm{F}\mathrm{I}\mathrm{N}}_2^{[\mathrm{\infty }]}$. We further extend our canonization theorem to arbitrary Borel maps $g:{\mathrm{F}\mathrm{I}\mathrm{N}}_2^{[\mathrm{\infty }]}\to \mathbb{R}$, establishing new canonical Ramsey theorems for Polish spaces. In particular, we provide a complete classification of the canonical functions on ${\mathrm{F}\mathrm{I}\mathrm{N}}_2^{[n]}$, together with an explicit formula for their number as a function of $n$.