# Sh:724

- Shelah, S. (2004).
*On nice equivalence relations on ^\lambda 2*. Arch. Math. Logic,**43**(1), 31–64. arXiv: math/0009064 DOI: 10.1007/s00153-003-0183-1 MR: 2036248 -
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.

- 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}, doi = {10.1007/s00153-003-0183-1}, mrclass = {03E47 (03E35 20K40)}, mrnumber = {2036248}, mrreviewer = {Andrzej Ros{\l}anowski}, doi = {10.1007/s00153-003-0183-1}, note = {\href{https://arxiv.org/abs/math/0009064}{arXiv: math/0009064}}, arxiv_number = {math/0009064} }