Representations of infinite tree-sets
J. Pascal Gollin, Jay Lilian Kneip
TL;DR
The paper addresses representing infinite tree sets as edge tree sets of graphs, identifying an obstruction: a chain of order type $\omega+1$. It develops a two-step program: first characterizing representable tree sets as regular tame tree sets and constructing the associated tree $T(\tau)$, then introducing tree-like spaces to capture all remaining cases and obtain a full representation theorem. The main contributions are (i) a precise characterization that regular tame tree sets are exactly the edge tree sets of (possibly infinite) graphs, (ii) a construction $T(\tau)$ realizing this correspondence, and (iii) a general representation theorem using tree-like spaces that represents every tree set and a characterisation of the corresponding spaces. This advances a unified framework for separations and tangles in graphs, with links to graph-like spaces and potential extensions to image analysis contexts.
Abstract
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ${ω+1}$. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space.