The Finite Length Property of the Rado Graph and Friends

TL;DR

This paper generalizes the finite length property to various infinite structures, including the Rado graph, by analyzing their automorphism orbits and structural approximations.

Contribution

It extends the finite length property to structures with finite orbit counts and Fra"issé limits with free amalgamation, including the Rado graph.

Findings

Proved the finite length property for the Rado graph.

Extended the property to structures approximated by finite substructures with few orbits.

Connected the property to function spaces and automata theory.

Abstract

An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length. Prior work showed that the countable pure set and the countable dense linear order without endpoints have this property. We generalise these results to (a) any structure approximated by finite substructures with few orbits, provided the field is of characteristic zero, and (b) any Fra\"iss\'e limit with free amalgamation in a finite vocabulary consisting of unary and binary relations, possibly expanded with a generic total order. As a special case, we deduce the finite length property of the Rado graph using both methods. We also describe some connections with function spaces, weighted register automata, and orbit-finite systems of linear equations.