On nice equivalence relations on 2^\lambda
  The main question here is the possible generalization of the following theorem on ``simple'' equivalence relation on 2^omega to higher cardinals.   Theorem: (1) Assume that:   (a) E is a Borel 2-place relation on 2^omega,   (b) E is an equivalence relation,   (c) if eta, nu in 2^omega and (exists ! n)(eta(n) not= nu(n)), then eta, nu are not E --equivalent.   Then there is a perfect subset of 2^omega of pairwise non E-equivalent members.   (2) Instead of ``E is Borel'', ``E is analytic (or even a Borel combination of analytic relations)'' is enough.   (3) If E is a Pi^1_2 relation which is an equivalence relation satisfying clauses (b)+(c) in V^Cohen, then the conclusion of (1) holds. 
Saharon Shelah