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Entangled Suslin lines and OGA

Carlos Martinez-Ranero, Lucas Polymeris

TL;DR

The paper proves the consistency of Open Graph Axiom (OGA) with the existence of a $2$-entangled Suslin line, by introducing the stronger notion of weakly bi-entangledness and developing an $E_S$-proper forcing framework that preserves Suslinity and entanglement through countable-support iterations. Starting from a supercompact cardinal, the authors execute a bookkeeping-guided iteration to force all instances of OGA while maintaining a $2$-entangled Suslin line, thus resolving questions by Carroy–Levine–Notaro and McKenney. The approach combines forcing to create weakly bi-entangled lines, preservation results for Suslin trees, and a Todorcevic-type forcing for OGA within a carefully controlled iteration. This yields a model where OGA holds and the $2$-entangled structure remains, with potential implications for the interaction between graph combinatorics and linear order topology. The work also raises questions about the necessity of large cardinals for such models and about potential purely ZFC implementations.

Abstract

We construct a model of the Open Graph Axiom (OGA) in which there is a 2-entangled Suslin line $S$. Consequently, in this model, there is a 2-entangled uncountable linear order, but no such order is separable. This resolves a problem posed by Carroy, Levine, and Notaro \cite{carroy2025} and answers a question from McKenney on MathOverflow \cite{Mckenney2014}.

Entangled Suslin lines and OGA

TL;DR

The paper proves the consistency of Open Graph Axiom (OGA) with the existence of a -entangled Suslin line, by introducing the stronger notion of weakly bi-entangledness and developing an -proper forcing framework that preserves Suslinity and entanglement through countable-support iterations. Starting from a supercompact cardinal, the authors execute a bookkeeping-guided iteration to force all instances of OGA while maintaining a -entangled Suslin line, thus resolving questions by Carroy–Levine–Notaro and McKenney. The approach combines forcing to create weakly bi-entangled lines, preservation results for Suslin trees, and a Todorcevic-type forcing for OGA within a carefully controlled iteration. This yields a model where OGA holds and the -entangled structure remains, with potential implications for the interaction between graph combinatorics and linear order topology. The work also raises questions about the necessity of large cardinals for such models and about potential purely ZFC implementations.

Abstract

We construct a model of the Open Graph Axiom (OGA) in which there is a 2-entangled Suslin line . Consequently, in this model, there is a 2-entangled uncountable linear order, but no such order is separable. This resolves a problem posed by Carroy, Levine, and Notaro \cite{carroy2025} and answers a question from McKenney on MathOverflow \cite{Mckenney2014}.

Paper Structure

This paper contains 14 sections, 27 theorems, 23 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Trees and lexicographic orderings
  4. Entangled linear orders
  5. The Open Graph Axiom
  6. Weakly bi-entangledness
  7. Forcing a weakly bi-entangled line
  8. Preserving 2-entangled Suslin lines
  9. Preserving Suslin trees
  10. Preserving weakly bi-entangled
  11. Forcing the Open Graph Axiom
  12. Preservation under countable support iterations.
  13. Consistency results
  14. Final remarks and open questions

Key Result

Proposition 2.6

Let $(S,<_{S},<_{\mathop{\mathrm{lex}}\nolimits})$ be a lexicographically ordered Suslin tree such that $(S,<_{\mathop{\mathrm{lex}}\nolimits})$ is a dense linear order. Then $(S,<_{\mathop{\mathrm{lex}}\nolimits})$ is $2$-entangled if and only if it is weakly $2$-entangled.

Theorems & Definitions (65)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Definition 3.1
  • ...and 55 more