## 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)

### Citations:

Zbl 1207.54044; Zbl 0569.03021; Zbl 0362.04001
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### References:

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