One of the initial motivations for the development of descriptive set theory (Borel-Baire-Lebesgue in Paris, Lusin-Egorov in Moscow) was to avoid the difficuilties of abstract set theory by focusing on sets of reals which have definitions of low complexity. In this talk I'll take a look at the extent to which this idea succeeds in the study of definable equivalence relations. An analytic equivalence relation can have countably-many (small), uncountably-many but not perfectly-many (medium), or perfectly-many classes (large); in the last case it can be either Borel or non-Borel.

The classes of an analytic equivalence relation can be countable (small) or contain a perfect set (large). For co-analytic equivalence relations they can also be uncountable with no perfect subset (medium). In either case a large class can either be Borel or non-Borel. I'll discuss the absoluteness/non-absoluteness of these notions as well as some related questions which connect to issues in the theory of class forcing.