The Honest Generalization of Silver's Dichotomy for Equivalence Relations (part 1)
A classic result in invariant descriptive set theory is Silver's dichotomy: every co-analytic equivalence relations on the reals either produce a countable quotient or there is a perfect set of inequivalent reals. Assuming the axiom of determinacy (AD), a similar theorem (uncountable Borel sets contain perfect sets) extends to arbitrary sets of reals, so it is not unreasonable to think that Silver's dichotomy might extend to arbitrary equivalence relations on the reals. In the 80s, Shelah proved a result that almost, but doesn't quite do this. AD implies that co-Suslin equivalence relations either produce well-ordered quotients (with a bound on the size given by the Suslinality) or there is a perfect set of inequivalent reals. We will sketch a proof of Shelah's result and discuss the strength and limitations of it.