CUNY Logic Workshop, New York, February 2010
Abstract: A (Borel) equivalence relation E is said to be hyperfinite iff it can be written as a countable increasing union of equivalence relations Fn, each of whose classes is finite. Although these are some of the simplest Borel equivalence relations, there are many open questions concerning their structure.
Recently, Stephen Jackson defined a combinatorial property of equivalence relations called Borel boundedness. Roughly speaking, it says that if a sequence of families of elements of the Baire space are indexed by $E$, then there is a Borel way to find a bound for each of them. It is easy to see that the hyperfinite relations are Borel bounded. We do not know whether any other equivalence relations have this property, but Jackson’s results imply that the more relations we can decide, then the more we know about the structure of the hyperfinite relations.
In this talk, I’ll carefully define all of the above concepts and then give a precise definition of Borel boundedness. Then, I’ll define several similar properties of equivalence relations which, like Borel boundedness, may yield information on the structure of hyperfinite equivalence relations.