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

Abstract

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