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

We consider the following dichotomy for $Σ^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $κ$: either all $R$-independent sets are of size at most $κ$, or there is a $κ$-perfect $R$-independent set. This dichotomy is the uncountable version of a result found in (W. Kubiś, 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 $\Diamond_κ$ and the set theoretical hypothesis $I^-(κ)$, which is the modification of the hypothesis $I(κ)$ suitable for limit cardinals. When $κ$ is inaccessible, or when $R$ is a closed binary relation, the assumption $\Diamond_κ$ is not needed. We obtain as a corollary the uncountable version of a result by G. Sági and the first author (Log. J. IGPL 20 (2012), no. 6:1064--1082) about the $κ$-sized models of a $Σ^1_1(L_{κ^+κ})$-sentence when considered up to isomorphism, or elementary embeddability, by elements of a $K_κ$ subset of ${}^κκ$. The role of elementary embeddings can be replaced by a more general notion that also includes embeddings, as well as the maps preserving $L_{λμ}$ for $ω\leqμ\leqλ\leqκ$ and the finite variable fragments of these logics.

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