Limit-closed Profiles

TL;DR

This paper extends tangle-tree theorems to infinite abstract separation systems by focusing on restrictions on tangles rather than the separation systems themselves, broadening applicability in graph theory.

Contribution

It introduces a new tangle-tree theorem for infinite separation systems without restrictions on the systems, only on the tangles, advancing the theoretical framework.

Findings

Established a tangle-tree theorem for infinite separation systems

Removed restrictions on the separation systems themselves

Applicable to a broader class of infinite graphs

Abstract

Tangle-tree theorems are an important tool in structural graph theory, and abstract separation systems are a very general setting in which tangle-tree theorems can still be formulated and proven. For infinite abstract separation systems, so far tangle-tree theorems have only been shown for special cases of separation systems, in particular when the separation system arises from a (locally finite) infinite graph. We present a tangle-tree theorem for infinite separation systems where we do not place restrictions on the separation system itself but on the tangles to be arranged in a tree.