A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

Abstract

We consider the following dichotomy for \(\Sigma^0_2\) finitary relations \(R\) on analytic subsets of the generalized Baire space for \(\kappa\): either all \(R\)-independent sets are of size at most \(\kappa\), or there is a \(\kappa\)-perfect \(R\)-independent set. This dichotomy is the uncountable version of a result found in (W. Kubi\'s, Proc. Amer. Math. Soc. 131 (2003), no 2.:619--623) and in (S. Shelah, Fund. Math. 159 (1999), no. 1:1--50). We prove that the above statement holds assuming \(\Diamond_\kappa\) and the set theoretical hypothesis \(I^-(\kappa)\), which is the modification of the hypothesis \(I(\kappa)\) suitable for limit cardinals. When \(\kappa\) is inaccessible, or when the relation \(R\) is closed, the assumption \(\Diamond_\kappa\) is not needed. We obtain as a corollary the uncountable version of a result by G. S\'agi and the first author (Log. J. IGPL 20 (2012), no. 6:1064--1082) about the \(\kappa\)-sized models of a \(\Sigma^1_1(L_{\kappa^+\kappa})\)-sentence when considered up to isomorphism, or elementary embeddability, by elements of a \(K_\kappa\) subset of \({}^\kappa\kappa\). The role of elementary embeddings can be replaced by a more general notion that also includes embeddings, as well as the maps preserving \(L_{\lambda\mu}\) for \(\omega\leq\mu\leq\lambda\leq\kappa\) and the finite variable fragments of these logics.