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 $R$ is a closed binary relation, 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.