The two‐dimensional version of the classical Mycielski theorem says that for every comeager or conull set there exists a perfect set such that. We consider a strengthening of this theorem by replacing a perfect square with a rectangle, where A and B are bodies of some types of trees with. In particular, we show that for every comeager Gδ set there exist a Miller tree and a uniformly perfect tree such that and that cannot be a Miller tree. In the (...) case of measure we show that for every subset F of of full measure there exists a uniformly perfect tree such that and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions from which we derive nonstandard proofs of Mycielski‐like theorems via the Shoenfield Absoluteness Theorem. (shrink)

We describe a list of canonical functions from 2 to ℝ such that every Borel measurable function from 2 to ℝ, on some superperfect rectangle, induces the same equivalence relation as some canonical function.

I describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relationsE, Fsuch that a natural proof of nonreducibility ofEtoFuses the independence of the Singular Cardinal Hypothesis at ℵω.