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

We consider the following dichotomy for ${\mathrm{\Sigma }}_2^0$ 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 ${\mathrm{◊}}_{\kappa }$ 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 ${\mathrm{◊}}_{\kappa }$ 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 ${\mathrm{\Sigma }}_1^1(L_{{\kappa }^{+}\kappa })$-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 $\omega \le \mu \le \lambda \le \kappa$ and the finite variable fragments of these logics.