A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals
Volume 238 / 2017
Abstract
We consider the following dichotomy for $ {\mathbf {\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. Kubiś, Proc. Amer. Math. Soc. 131 (2003), 619–623] and in [S. Shelah, Fund. Math. 159 (1999), 1–50]. We prove that the above statement holds if we assume $\Diamond _\kappa $ and the set-theoretical hypothesis $ {\mathrm {I}^-(\kappa )}$, which is the modification of the hypothesis $ {\mathrm {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. Sági and the first author [Logic J. IGPL 20 (2012), 1064–1082] about the $\kappa $-sized models of a $ {\mathbf {\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 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 finite variable fragments of these logics.
