On nice equivalence relations on ${}^\lambda 2$

by Shelah. [Sh:724]
Archive for Math Logic, 2004
The main question here is the possible generalization of the following theorem on ``simple'' equivalence relation on {}^omega 2 to higher cardinals. Theorem: (1) Assume that (a) E is a Borel 2-place relation on {}^omega 2, (b) E is an equivalence relation, (c) if eta, nu in {}^omega 2 and (exists !n)(eta (n) not= nu (n)), then eta, nu are not E --equivalent. Then there is a perfect subset of {}^omega 2 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.

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