Borel Reductions and Cub Games in Generalized Descriptive Set Theory
Vadim Kulikov
TL;DR
The paper proves that for an uncountable regular κ with $κ^{<κ}=κ$, the poset $\langle \mathcal{P}(\kappa),\subset_{\operatorname{NS}(\lambda)}\rangle$ can be embedded into the Borel equivalence relations on $2^{\kappa}$ under Borel reducibility, with the embedding placing the image strictly between $\operatorname{id}_{2^{\kappa}}$ and $E_0$. This embedding extends known ω-case results to generalized descriptive set theory and relies on GC$_{\lambda}$-characterization, cub games, and square- and non-reflection principles. Under several set-theoretic hypotheses (e.g., $\kappa=\omega_1$ or $\kappa=\lambda^{+}$ with $\square_{\lambda}$), the stronger NS embedding $\subset_{\operatorname{NS}}$ is attainable as well, and the paper derives numerous corollaries. The authors also construct long chains of Borel equivalence relations of length $\kappa^{+}$, illustrating the rich order structure of $\mathcal{E}^{B}_{\kappa}$ and connecting these descriptive-set-theoretic objects to combinatorial set theory notions like stationary and non-stationary sets.
Abstract
It is shown that the power set of $κ$ ordered by the subset relation modulo various versions of the non-stationary deal can be embedded into the partial order of Borel equivalence relations on $2^κ$ under Borel reducibility. Here $κ$ is uncountable regular cardinal with $κ^{<κ} = κ$.