Unbounded Class of Orbit Equivalence Relations

In their paper titled "The Classification of Hypersmooth Borel Equivalence Relations" Alexander Kechris and Alain Louveau quote the following (Theorem 5.2 in the article) as "Harrington, unpublished":

There is a nonempty class $U$ of Borel equivalence relations which is unbounded, and every $F \in U$ is induced by a Borel action of a Polish group (so in particular is idealistic).

I have been unable to find any proof of that, help would be greatly appreciated.

The only idea (which probably leads nowhere anyway) I had to get higher and higher orbit equivalence relations was to apply the FS-jump to some orbit equivalence relations, since those again can be shown to be induced by Polish group actions. But that's not sufficient for the theorem by any means, so I'm not sure it makes sense to mention this here.

flag	asked May 5 2012 at 10:46 ftonti 182●7

The various iterates $T_{\alpha}$ for $\alpha < \omega_{1}$ of the $FS$-jump are indeed unbounded within the class of Borel equivalence relations. And this is all that is needed for the application in Kechris-Louveau. – Simon Thomas May 5 2012 at 12:05

For example, see: MR1624736 (2000a:03081) Hjorth, Greg(1-UCLA); Kechris, Alexander S.(1-CAIT); Louveau, Alain(F-PARIS6-E) Borel equivalence relations induced by actions of the symmetric group. (English summary) Ann. Pure Appl. Logic 92 (1998), no. 1, 63–112. – Simon Thomas May 5 2012 at 12:09

Thank you very much, that is very helpful! By the way, you could post this as an answer, if you wanted. – ftonti May 5 2012 at 15:03

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