Algorithmic randomness and Ramsey properties of countable homogeneous structures

TL;DR

This paper explores the connection between algorithmic randomness and the symmetries of countable homogeneous structures, focusing on Ramsey properties and their relation to amenable subgroups of the infinite symmetric group.

Contribution

It establishes a novel link between Ramsey Fra"issé order classes and algorithmic randomness within the context of symmetric groups.

Findings

Identifies relationships between Ramsey properties and algorithmic randomness.

Characterizes closed amenable subgroups of the symmetric group.

Provides insights into the structure of symmetries in countable homogeneous structures.

Abstract

We study, in the context of algorithmic randomness, the closed amenable subgroups of the symmetric group $S_\infty$ of a countable set. In this paper we address this problem by investigating a link between the symmetries associated with Ramsey Fra\"iss\'e order classes and algorithmic randomness.