Indiscernible pairs of countable sets of reals at a given projective level

Abstract

Using an invariant modification of Jensen's "minimal ${\Pi }_2^1$ singleton" forcing, we define a model of ZFC, in which, for a given $n\ge 2$, there exists a lightface ${\Pi }_n^1$ unordered pair of non-OD (hence, OD-indiscernible) countable sets of reals, but there is no ${\Sigma }_n^1$ unordered pairs of this kind.