Canonical equivalence relations on fronts on $FI{N}_k$

We prove that for every equivalence relation on a barrier on the space $⟨FI{N}_k^{[\mathrm{\infty }]},\le ,r⟩$, for any $k$, there exists $Y\in FIN_k^{[\infty]}$sothattherestrictionofthecoloringon$\langle Y\rangle$ is canonical.