We show that any symmetric, Baire measurable function from the complement of $E_0$ to a finite set is constant on an $E_0$-nonsmooth square. A simultaneous generalization of Galvin's theorem that Baire measurable colorings admit perfect homogeneous sets and the Kanovei-Zapletal theorem canonizing Borel equivalence relations on $E_0$-nonsmooth sets, this result is proved by relating $E_0$-nonsmooth sets to embeddings of the complete binary tree into itself and appealing to a version of Hindman's theorem on the complete binary tree. We also establish several canonization theorems which follow from the main result.
"Canonizing relations on nonsmooth sets." J. Symbolic Logic 78 (1) 101 - 112, March 2013. https://doi.org/10.2178/jsl.7801070