On the Laczkovich–Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence ( A n ) of analytic subsets of a Polish space X, if lim sup n ∈ H A n is uncountable for every H ∈ [ N ] ω then ⋂ n ∈ G A n is uncountable for some G ∈ [ N ] ω . This fact, by our definition, means that the σ-ideal [ X ] ⩽ ω has property (LK). We prove that every σ-ideal generated by X / E has property (LK), for an equivalence relation E ⊂ X 2 of type F σ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.