On the Laczkovich–Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then ⋂n∈GAn is uncountable for some G∈ω[N]. 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⊂X2 of type Fσ 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.