Counting the back-and-forth types

Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many n-back-and-forth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n− 1)st jump of some structure in K. In the former case we show that there is a countable set of infinitary Πn relations that captures all of the Πn information about the structures in K. In most cases where there are countably many n-back-and-forth equivalence classes, there is a computable description of them. We will show how to use this computable description to get a complete set of computably infinitary Πn formulas. This will allow us to completely characterize the relatively intrinsically Σn+1 relations in the computable structures of K, and to prove that no Turing degree can be coded by the (n − 1)st jump of any structure in K unless that degree is already below 0(n−1).