On the classification of vertex-transitive structures

TL;DR

This paper investigates the complexity of classifying various countable vertex-transitive structures, revealing that some classifications are highly complex and others have specific known complexities.

Contribution

It establishes the Borel completeness for countable vertex-transitive digraphs and partial orders, and determines the complexity levels for linear orders and tournaments.

Findings

Classification of countable vertex-transitive digraphs is Borel complete.

Classification of countable vertex-transitive partial orders is Borel complete.

Classification of vertex-transitive countable tournaments exceeds $E_0$ in complexity.

Abstract

We consider the classification problem for several classes of countable structures which are "vertex-transitive", meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above $E_0$ in complexity.