Graphings of arithmetical equivalence relations

This paper studies when an arithmetical equivalence relation E can be realized as the connectedness relation of a graph G which is simpler to define than E. Several examples of such equivalence relations are established. In particular, it is proved that the Σ⁰₃ relation of computable isomorphism of structures on in a computable first-order language is Π⁰₂-graphable, i.e., is the connectedness relation of a Π⁰₂ graph. Graphings of Friedman-Stanley jumps are studied, including an arithmetical construction of a graphing of the Friedman-Stanley jump of E from a graphing of E.