Graphings of arithmetical equivalence relations
Tyler Arant
TL;DR
The paper develops a comprehensive framework for realizing arithmetical equivalence relations as graph-connectedness relations of graphs with simple, low-arithmetical definitions. It proves broad $\Pi^0_2$-graphability results with diameter 2 for key relations (e.g., $E_0$, $\equiv_T$, index relations, computable isomorphism) and shows that Friedman–Stanley jumps preserve graphability: from a finite-diameter graphing of $E$ one can define a $\forall^\mathbb{N}$-graphing of $E^+$ with diameter at most $\max(2,\ell)$. The paper also establishes robust closure properties under products and invariant reductions, and provides optimal negative results (notably the Shinko–Weilacher theorem and related non-graphability examples) that delineate the boundaries of the approach. Together, these results yield a coherent picture of how arithmetical structure can be captured by simpler, effectively definable graphs, with wide implications for computable structure theory and descriptive set theory. The methods hinge on explicit, computable encodings (e.g., shuffles, coding relations, and witnessing predicates) that produce concrete graphings of diameter 2 across a broad class of relations and their FS jumps.
Abstract
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 $Σ^0_3$ relation of computable isomorphism of structures on $\N$ in a computable first-order language is $Π^0_2$-graphable, i.e., is the connectedness relation of a $Π^0_2$ 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 $E$.