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

Abstract

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 Rindependent 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 ♦ 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 ♦ 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 κ κ. 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 ≤ µ ≤ λ ≤ κ and the finite variable fragments of these logics.

The open dihypergraph dichotomy for generalized Baire spaces and its applications

“…( This strengthens a joint result of Väänänen and the second-listed author [SzV17] showing that DK κ (Σ 1 1 (κ)) holds in Col(κ, <λ)-generic extensions for any measurable cardinal λ > κ. 154 It is open whether one can obtain DK κ (Σ 1 1 ) directly from ODD κ κ ( κ κ) by constructing a suitable dihypergraph.…”

“…This strengthens two results of Väänänen [Vää91, Theorem 1 & Theorem 4] showing that the principle I(ω) that is equiconsistent with a measurable cardinal 148 implies (a) II wins V ω 1 (X) for all subsets X of ω 1 ω 1 of size >ω 1 149 and (b) CB 2 ω 1 (X) for all closed subsets X of ω 1 ω 1 . 150 Using the previous corollary and [Szir18, Corollary 4.35], we can lower the consistency strength of the dichotomy on the size of complete subhypergraphs (cliques) of finite dimensional Π 0 2 (κ) dihypergraphs on Σ 1 1 (κ) sets studied in [SzV17] and its generalization to families of hypergraphs. Given a family H of dihypergraphs on a set X, we say that Y ⊆ X is an H-clique if K d H Y ⊆ H for each H ∈ H, where d H denotes the arity of H.…”

“…The latter extends a result of Väänänen [Vää91] from closed sets to arbitrary subsets of κ κ. Moreover, it lowers the upper bound for the consistency strength of a dichotomy studied in [SzV17] from a measurable to a Mahlo cardinal.…”