Finitely Bounded Homogeneity Turned Inside-Out (2108.00452v10)

Abstract: The classification problem for countable finitely bounded homogeneous structures is notoriously difficult, with only a handful of published partial classification results, e.g., for directed graphs. We introduce the Inside-Out correspondence, which links the classification problem, viewed as a computational decision problem, to the problem of testing the embeddability between reducts of countable finitely bounded homogeneous structures. On the one hand, the correspondence enables polynomial-time reductions from various decision problems that can be represented within the embeddability problem, e.g., the double-exponential square tiling problem. This leads to a new lower bound for the complexity of the classification problem: $\mathsf{2NEXPTIME}$-hardness. On the other hand, it also follows from the Inside-Out correspondence that the classification (decision) problem is effectively reducible to the (search) problem of finding a finitely bounded Ramsey expansion of a countable finitely bounded homogeneous structure. We subsequently prove that the closely related problem of homogenizability is already undecidable.