The finite length property of the Rado Graph and friends

Conference item

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.

Authors: Yang, J; Bojańczyk, M; Klin, B

• Publication status: Accepted
• Peer review status: Peer reviewed
• Publisher: Dagstuhl Publishing
• Host title: Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik
• Pages: 6:1–6:27
• Article number: 6
• Acceptance date: 2026-04-16
• Event title: 41st Annual Symposium on Logic in Computer Science (LICS)
• Event location: Lisbon, Portugal
• Event website: https://lics.siglog.org/lics26/
• Event start date: 2026-07-20
• Event end date: 2026-07-23
• DOI: 10.4230/LIPIcs.LICS.2026.6
• Language: English
• Keywords: Rado Graph; oligomorphic structure; orbit-finite set; orbit-finitely spanned vector space; equivariant subspace; finite length