### 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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