Model Theory of Homogeneous D-sets
Felipe Estrada, John Goodrick
TL;DR
This work analyzes treelike relational structures called $D$-sets and their colored variants to understand model-theoretic tameness. By linking $D$-relations $D(wx;yz)$ to a tree-structure through splittings and sectors, it provides a tree-based dictionary for finite and infinite $D$-sets, including a complete characterization of ultrahomogeneous colored $D$-sets and a detailed account of indiscernibles, distality, and dp-minimality. The main achievements include a precise ultrahomogeneity criterion for countable dense proper colored $D$-sets, a finite-tree representation for finite $D$-sets with a node/edge splitting correspondence, and a hull-based framework that yields distality and dp-minimality results when quantifier elimination is available. The results illuminate how treelike combinatorics control model-theoretic tameness in these structures, with potential implications for related permutation groups and tame geometry in logical frameworks.
Abstract
We explore several model-theoretic aspects of D-sets, which were studied in detail by Adeleke and Neumann. We characterize ultrahomogeneity in the class of colored D-sets and classify unbounded order-indiscernible sequences in such structures. We use these results to provide a characterization of distal colored D-sets and prove that all colored D-sets with quantifier elimination are dp-minimal.