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The Finite Length Property of the Rado Graph and Friends

Authors Jingjie Yang , Mikołaj Bojańczyk , Bartek Klin

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LIPIcs.LICS.2026.82.pdf
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ACM Subject Classification
  • Theory of computation → Logic
  • Mathematics of computing → Random graphs
  • Theory of computation → Automata over infinite objects
Keywords
  • Rado graph
  • oligomorphic structure
  • orbit-finite set
  • orbit-finitely spanned vector space
  • equivariant subspace
  • finite length

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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.

Cite As Get BibTex

Jingjie Yang, Mikołaj Bojańczyk, and Bartek Klin. The Finite Length Property of the Rado Graph and Friends. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 82:1-82:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.82

Author Details

Jingjie Yang
  • University of Oxford, UK
Mikołaj Bojańczyk
  • University of Warsaw, Poland
Bartek Klin
  • University of Oxford, UK

Funding

Mikołaj Bojańczyk was supported by the Polish National Science Centre (NCN) grant "Polynomial finite state computation" (2022/46/A/ST6/00072). Jingjie Yang was partially supported by the Polish National Science Centre (NCN) grant "Linear algebra in orbit-finite dimension" (2022/45/N/ST6/03242).

Acknowledgements

We thank David M. Evans, Arka Ghosh, and Ehud Hrushovski for helpful discussions. We are very grateful to the anonymous reviewers for their insightful comments.

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