Borel Reductions and Cub Games in Generalized Descriptive Set Theory

It is shown that the power set of $\kappa$ 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 $2^\kappa$ under Borel reducibility. Here $\kappa$ is uncountable regular cardinal with $\kappa^{<\kappa} = \kappa$.