The {L}aczkovich—{K}omjáth property for coanalytic equivalence relations

Abstract

Let E be a coanalytic equivalence relation on a Polish space X and (A_n)_{n ∈ ω} a sequence of analytic subsets of X. We prove that if lim sup_{n ∈ K} A_n meets uncountably many E-equivalence classes for every K ∈ [ω]^ω, then there exists a K ∈ [ω]^ω such that ⋂_{n ∈ K} A_n contains a perfect set of pairwise E-inequivalent elements.