Dependent choice and iterated beta_n-model reflection

In this paper, we consider conservation results for iterated $β_n$-model reflection principles. Using these principles, we characterize the $Π^1_{m+2}$-, $Σ^1_{m+2}$- and $\mathsf{B}(Π^1_{m+2})$-consequences of $Σ^1_{n+1}$-$\mathsf{DC}_0$ and $Σ^1_{n+1}$-$\mathsf{TDC}_0$ for $m \leq n$. Here, $\mathsf{B}(Π^1_{m+2})$ denotes the set of Boolean combinations of $Π^1_{m+2}$-formulas, and $\mathsf{TDC}$ stands for transfinite $\mathsf{DC}$. We also show that, in each case, the stronger theory has a super-exponential speed-up over the corresponding conservative subtheory.