# Sh:724

• Shelah, S. (2004). On nice equivalence relations on ^\lambda 2. Arch. Math. Logic, 43(1), 31–64.
• Abstract:
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)\neq \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^{\rm Cohen}, then the conclusion of (1) holds.

• Current version: 2003-07-02_10 (49p) published version (34p)
Bib entry
@article{Sh:724,
author = {Shelah, Saharon},
title = {{On nice equivalence relations on $^\lambda 2$}},
journal = {Arch. Math. Logic},
fjournal = {Archive for Mathematical Logic},
volume = {43},
number = {1},
year = {2004},
pages = {31--64},
issn = {0933-5846},
mrnumber = {2036248},
mrclass = {03E47 (03E35 20K40)},
doi = {10.1007/s00153-003-0183-1},
note = {\href{https://arxiv.org/abs/math/0009064}{arXiv: math/0009064}},
arxiv_number = {math/0009064}
}