Canonizing relations on nonsmooth sets

We show that any symmetric, Baire measurable function from the complement of E0 to a finite set is constant on an E0-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 E0-nonsmooth sets, this result is proved by relating E0-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. §0. Introduction. While it is well known that many Ramsey-style partition properties fail at uncountable cardinals, it is perhaps surprising that their descriptive analogs often hold. For example, using a wellordering of the reals it is easy to build a two-coloring of pairs of real numbers which admits no uncountable homogeneous set. If we restrict our attention to Baire measurable colorings of pairs, however, Galvin’s theorem (see [2, Theorem 19.7]) ensures that we may always find a homogeneous perfect subset. At first glance, this may seem as far as one could hope to push things, as the classical framework for descriptive set theory lies within the confines of Polish spaces, whose cardinalities are bounded above by that of the continuum. However, in the descriptive context we must also change our outlook on cardinality. For example, given two countable Borel equivalence relations E and F on Polish spaces X and Y , there may be no Borel function φ : X → Y such that x0 E x1 ⇔ φ(x0) F φ(x1) (such a function is a reduction of E to F ) nor a Borel function ψ : Y → X with the analogous property. In such a situation, there is no Borel way of comparing the quotient spaces X/E and Y/F , even though of course each has the cardinality of the continuum. In that sense, the “Borel cardinality” of quotient spaces can be very complicated. As expected, the list of the “Borel cardinal numbers” begins 0, 1, 2, . . . ,א0, c. Remarkably, there is a cardinal successor of the continuum, namely 2/E0, where E0 is the equivalence relation of eventual agreement of binary strings. This is the celebrated Harrington-Kechris-Louveau generalization of the Glimm-Effros dichotomy (see [1]): if a Borel equivalence relation E does not Borel reduce to equality of reals, then there is a Borel reduction of E0 to E. One naturally wonders about the Ramsey-theoretic properties this next Borel cardinal might possess. Supported in part by Marie Curie grant no. 249167 from the European Union.