Canonizing relations on nonsmooth sets

Abstract

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.