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

TL;DR

This paper establishes a dichotomy for $oldsymbol{ ext{Sigma}}^0_2$ relations on analytic subsets of the generalized Baire space at uncountable cardinals, linking set-theoretic assumptions with the existence of large independent sets.

Contribution

It extends a known dichotomy from countable to uncountable settings, under specific set-theoretic hypotheses, and applies it to models and elementary embeddability at uncountable cardinals.

Findings

The dichotomy holds assuming $ riangle_oldsymbol{ ext{Diamond}}_oldsymbol{ ext{κ}}$ and $I^-(oldsymbol{ ext{κ}})$.

When $oldsymbol{ ext{κ}}$ is inaccessible or $R$ is closed, no additional assumptions are needed.

Corollaries include uncountable versions of results on models of $oldsymbol{ ext{Σ}}^1_1$ sentences and elementary embeddability.

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.