On large cardinals and generalized Baire spaces

Abstract

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $κ$. We show the consistency of $E^{λ^{++},λ^{++}}_{λ\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_λ$ in the space $(λ^{++})^{λ^{++}}$, being continuously reducible to $E^{2,λ^{++}}_{λ^+\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_{λ^+}$ in the space $2^{λ^{++}}$. Then we show the consistency of $E^{2,κ}_{reg}$, the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space $2^κ$, being $Σ_1^1$-complete. We finish by showing, for $Π_2^1$-indescribable $κ$, that the isomorphism relation between dense linear orders of cardinality $κ$ is $Σ_1^1$-complete.