Largest \(E\)-thin, \(E\)-invariant sets below \(\Delta_3^1\) (Q1882565)

Let \(E\) be an equivalence relation on a Polish space \(X\). A subset of \(X\) is \(E\)-thin if it does not contain a perfect set of pairwise inequivalent elements, it is \(E\)-invariant if it contains every equivalence class it intersects. The paper deals with the following question: if \(\Gamma\) and \(\Gamma'\) are point classes, is it true that for every equivalence class in \(\Gamma\) there exists a largest \(E\)-thin, \(E\)-invariant set belonging to \(\Gamma'\)? In unpublished notes dating from the late 1970's Kechris answered affirmatively the question when \(\Gamma = \Delta^1_1\) and \(\Gamma' = \Pi^1_1\), and when \(\Gamma = \Pi^1_1\) and \(\Gamma' = \Sigma^1_2\). Assuming \(0^\#\), Hjorth gave a positive answer for \(\Gamma = \Sigma^1_1\) and \(\Gamma' = \Sigma^1_2\). The main result of the paper is a negative answer to the above question when \(\Gamma = \Gamma' = \Pi^1_1\). The proof of this result yields negative answers in a few more cases, such as \(\Gamma = \Sigma^1_1\) and \(\Gamma' = \Pi^1_1\), and \(\Gamma = \Pi^1_2\) and \(\Gamma' = \Sigma^1_2\). Therefore the only open problems for projective classes below \(\Delta^1_3\) are \(\Gamma\) either \(\Delta^1_2\) or \(\Sigma^1_2\) and \(\Gamma' = \Sigma^1_2\). The main result is proved by a careful construction by effective induction.