The Finite Length Property of the Rado Graph and Friends

2026-05-20

AI Summary

This study investigates whether infinite structures over a given field possess the finite length property—namely, whether the lengths of equivariant subspace chains in their finite powers are uniformly bounded. The authors extend this property to broader classes of infinite structures through two general frameworks: first, structures in characteristic zero fields approximable by finite substructures with few orbits; and second, Fraïssé limits with free amalgamation (optionally equipped with a generic linear order). By integrating model theory, equivariant linear algebra, and orbit analysis, the work provides two independent proofs of the finite length property for the Rado graph and uncovers deep connections to function spaces, weighted register automata, and orbit-finite systems of linear equations, thereby establishing a theoretical bridge across mathematical logic and computational models.

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 Fraïssé 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.

Problem

• finite length property
• Rado graph
• Fraïssé limit
• equivariant subspaces
• orbit-finite

Innovation

• finite length property
• Rado graph
• Fraïssé limit
• free amalgamation
• equivariant subspaces