Flows of linear orders on sparse graphs

TL;DR

This paper studies the topological dynamics of automorphism groups of a specific sparse graph from Hrushovski construction, revealing that all minimal subflows have meagre orbits, thus addressing a question in the field.

Contribution

It provides new insights into the topological dynamics of automorphism groups of sparse graphs, specifically showing all orbits in minimal subflows are meagre.

Findings

All orbits in minimal subflows are meagre.

Addresses a question of Tsankov on sparse graphs.

Connects to prior work by Evans, Hubicka, and Nesetril.

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.