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

We consider the following dichotomy for ∑⁰₂ finitary relations R on analytic subsets of the generalized Baire space for k: either all R-independent sets are of size at most k, or there is a k-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 ◊_k and the set-theoretical hypothesis I^-(k), which is the modification of the hypothesis I(k) suitable for limit cardinals. When K is inaccessible, or when R is a closed binary relation, the assumption ◊_k 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 k-sized models of a ∑¹₁(L_k⁺k)-sentence when considered up to isomorphism, or elementary embeddability, by elements of a Kk subset of k_K. The elementary embeddings can be replaced by a more general notion that also includes embeddings, as well as the maps preserving L_λμ for ω≤μ≤λ≤κ and finite variable fragments of these logics.