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Trees of tangles in infinite separation systems

Christian Elbracht, Jay Lilian Kneip, Maximilian Teegen

TL;DR

The paper advances infinite tangles theory by extending the finite splinter lemma to infinite separation systems through two complementary routes. The profinite approach uses inverse limits of finite universes to obtain a canonical closed nested set of separations that distinguishes a broad class of profiles, yielding canonical trees of tree-decompositions and connections to ends and ultrafilter tangles. The thin splinter lemma provides a non-profinite, canonical framework based on a crossing-number thinning to produce separators and decompositions in general graphs, including locally finite and infinite-degree cases. Together, these methods unify infinite-tree-of-tangles results for graphs and abstract separation systems, producing canonical, isomorphism-invariant structures such as nested separator sets and tree-of-tree-decompositions that distinguish principal and robust profiles. The results have significant implications for understanding the cohesion structure and decompositions of infinite graphs, offering canonical, structurally robust tools aligned with ends, tangles, and profile theory.

Abstract

We present infinite analogues of our splinter lemma from [Trees of tangles in abstract separation systems, arXiv:1909.09030]. From these we derive several tree-of-tangles-type theorems for infinite graphs and infinite abstract separation systems.

Trees of tangles in infinite separation systems

TL;DR

The paper advances infinite tangles theory by extending the finite splinter lemma to infinite separation systems through two complementary routes. The profinite approach uses inverse limits of finite universes to obtain a canonical closed nested set of separations that distinguishes a broad class of profiles, yielding canonical trees of tree-decompositions and connections to ends and ultrafilter tangles. The thin splinter lemma provides a non-profinite, canonical framework based on a crossing-number thinning to produce separators and decompositions in general graphs, including locally finite and infinite-degree cases. Together, these methods unify infinite-tree-of-tangles results for graphs and abstract separation systems, producing canonical, isomorphism-invariant structures such as nested separator sets and tree-of-tree-decompositions that distinguish principal and robust profiles. The results have significant implications for understanding the cohesion structure and decompositions of infinite graphs, offering canonical, structurally robust tools aligned with ends, tangles, and profile theory.

Abstract

We present infinite analogues of our splinter lemma from [Trees of tangles in abstract separation systems, arXiv:1909.09030]. From these we derive several tree-of-tangles-type theorems for infinite graphs and infinite abstract separation systems.

Paper Structure

This paper contains 13 sections, 45 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Terminologyandbasicfacts
  3. Theprofinitesplinterlemma
  4. Introductiontoprofiniteuniverses
  5. Statementandproofoftheprofinitesplinterlemma
  6. Applicationoftheprofinitesplinterlemma
  7. Thethinsplinterlemma
  8. Applicationsofthethinsplinterlemma
  9. Locallyfinitegraphs
  10. Graphswithverticesofinfinitedegree
  11. Nestedsetsofseparations
  12. Canonicaltreesoftree-decompositions
  13. Correctness

Key Result

Theorem 1.1

Let$U$beauniverseofseparationsand$(\cA_i)_{i\le n}$afamilyofsubsetsof $U.$If$(\cA_i)_{i\le n}$splintersthenwecanpickanelement$a_i$fromeach$\cA_i$sothat$\{a_1,\dots,a_n\}$isnested.

Figures (1)

  • Figure 1: Alocallyfinitegraphwherenotree-decompositiondistinguishesalltherobustregularboundedprofilesefficiently.Thegreenseparatoristheoneoftheonlyseparationwhichefficientlydistinguishestheprofileinducedbythe $K^{64}$fromtheprofileinducedbythe $K^{128}$.

Theorems & Definitions (80)

  • Theorem 1.1: Splinterlemma,FiniteSplinters
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: carmesinhalinconj*Theorem5.12
  • Theorem 1.5: carmesin2020canonical*Remark8.3
  • Lemma 2.1: ProfileDuality
  • Lemma 2.2: halin1991lattices*2.4
  • Lemma 2.3: ProfilesNew*Lemma 2.1
  • Lemma 2.4
  • proof
  • ...and 70 more