Flows of linear orders on sparse graphs

TL;DR

This work investigates the topological dynamics of the automorphism group of the sparse Hrushovski limit $M_1$ by analyzing the flow of linear orders LO$(M_1)$. It demonstrates that every minimal Aut$(M_1)$-flow inside LO$(M_1)$ has all orbits meagre, using a Ramsey expansion of $\mathcal{D}_1$ via admissible orders to force a failure of the weak amalgamation property for the associated age. Consequently, LO$(M_1)$ is not a minimal flow and the universal minimal flow for Aut$(M_1)$ remains non-metrisable, providing a partial answer to Tsankov’s question on metrisable minimal flows in this sparse-graph setting. The results contribute to understanding non-tame topological dynamics for Hrushovski-type sparse graphs and illustrate the robustness of meagre-orbit phenomena beyond the previously studied orientations, linking Ramsey-theoretic methods to dynamical properties of automorphism groups.

Abstract

We consider the topological dynamics of the automorphism group of a particular sparse graph M_1 resulting from an ab initio Hrushovski construction. We show that minimal subflows of the flow of linear orders on M_1 have all orbits meagre, partially answering a question of Tsankov regarding results of Evans, Hubicka and Nesetril on the topological dynamics of automorphism groups of sparse graphs.

Figures (2)

• Figure 1: The oriented graph $D_T$.
• Figure 2: The ordered graph $C_T^\prec$ with witness vertices indicated on one vertex.

Theorems & Definitions (30)

• Theorem : BMT17
• Theorem : KPT05
• Theorem : KPT05, NVT13, Zuc16
• Theorem : EHN19
• Theorem : EHN19
• Theorem \ref{M1meagre}
• proof
• Definition 3.1
• Definition 3.2
• Proposition 3.3