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## Fundamenta Mathematicae

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

### Tom 238 / 2017

Fundamenta Mathematicae 238 (2017), 53-78 MSC: Primary 03E15; Secondary 03C45, 03C57, 03E55. DOI: 10.4064/fm130-9-2016 Opublikowany online: 27 December 2016

#### Streszczenie

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.

#### Autorzy

• Dorottya SzirákiAlfréd Rényi Institute of Mathematics
Reáltanoda u. 13–15
H-1053 Budapest, Hungary
and
Department of Mathematics
and its Applications
Central European University
Nádor u. 9
H-1051 Budapest, Hungary
e-mail
• Jouko VäänänenDepartment of Mathematics and Statistics
Gustaf Hällströmin katu 2b
FI-00014 University of Helsinki, Finland
and
Institute for Logic, Language and Computation
University of Amsterdam
1090 GE Amsterdam, Netherlands
e-mail

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