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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 versionof a result found in (W. Kubi\'s, Proc. Amer. Math. Soc. 131 (2003), no2.:619--623) and in (S. Shelah, Fund. Math. 159 (1999), no. 1:1--50). We provethat the above statement holds assuming $\Diamond_\kappa$ and the settheoretical hypothesis $I^-(\kappa)$, which is the modification of thehypothesis $I(\kappa)$ suitable for limit cardinals. When $\kappa$ isinaccessible, 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 andthe 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 whenconsidered up to isomorphism, or elementary embeddability, by elements of a$K_\kappa$ subset of ${}^\kappa\kappa$. The role of elementary embeddings canbe replaced by a more general notion that also includes embeddings, as well asthe maps preserving $L_{\lambda\mu}$ for $\omega\leq\mu\leq\lambda\leq\kappa$and the finite variable fragments of these logics.

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