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The Laczkovich-Komjáth property for coanalytic equivalence relations. (English) Zbl 1227.03065

An equivalence relation \(E\) on a Polish space \(X\) is said to have the Laczkovich-Komjáth property if for every sequence \(A_n\) of analytic subsets of \(X\) such that for every \(K\in[\omega]^\omega\) the set \(\limsup_{n\in K}A_n\) meets uncountably many \(E\)-equivalence classes there is \(L\in[\omega]^\omega\) such that \(\bigcap_{n\in L}A_n\) contains a perfect set of pairwise \(E\)-inequivalent elements. This terminology was introduced in [M. Balcerzak and S. Głąb, “On the Laczkovich-Komjáth property of sigma-ideals”, Topology Appl. 157, No. 2, 319–326 (2010; Zbl 1207.54044)]. Earlier, P. Komjáth [“On the limit superior of analytic sets”, Anal. Math. 10, 283–293 (1984; Zbl 0569.03021)] showed that the identity relation has the Laczkovich-Komjáth property, improving on an earlier result of M. Laczkovich [“On the limit superior of sequences of sets”, Anal. Math. 3, 199–206 (1977; Zbl 0362.04001)], who showed this in case the sets \(A_n\) in the above definition are Borel. Balcerzak and Głąb showed that every \(F_\sigma\) equivalence relation has the Laczkovich-Komjáth property.
In this paper, the authors show that every coanalytic equivalence relation has the Laczkovich-Komjáth property. In addition, the paper establishes the parametric version of the Laczkovich-Komjáth property, also considered by Balcerzak and Głąb.

MSC:

03E15 Descriptive set theory
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
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References:

[1] DOI: 10.1016/0003-4843(80)90002-9 · Zbl 0517.03018
[2] Descriptive set theory (1980) · Zbl 0433.03025
[3] Analytic sets (1980)
[4] Invariant descriptive set theory (2009) · Zbl 1154.03025
[5] DOI: 10.1007/BF01904778 · Zbl 0569.03021
[6] DOI: 10.1090/S0894-0347-1990-1057041-5
[7] DOI: 10.1090/S0002-9947-07-04243-2 · Zbl 1144.54025
[8] DOI: 10.1007/BF02297692 · Zbl 0362.04001
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