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Formalising Szemerédi’s Regularity Lemma in Lean

Authors Yaël Dillies , Bhavik Mehta

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Author Details

Yaël Dillies
  • Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK
Bhavik Mehta
  • Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK

Funding

  • Mehta, Bhavik: Cantab Capital Institute for the Mathematics of Information

Acknowledgements

We want to thank Anne Baanen, Thomas Bloom, Kevin Buzzard, Timothy Gowers, Heather Macbeth, the mathlib maintainers and the anonymous reviewers for useful comments on previous versions of this paper. We are grateful to the mathlib community, whose work has been invaluable for the prerequisites of this project.

Cite As Get BibTex

Yaël Dillies and Bhavik Mehta. Formalising Szemerédi’s Regularity Lemma in Lean. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITP.2022.9

Abstract

Szemerédi’s Regularity Lemma is a fundamental result in graph theory with extensive applications to combinatorics and number theory. In essence, it says that all graphs can be approximated by well-behaved unions of random bipartite graphs. We present a formalisation in the Lean theorem prover of a strong version of this lemma in which each part of the union must be approximately the same size. This stronger version has not been formalised previously in any theorem prover. Our proof closely follows the pen-and-paper method, allowing our formalisation to provide an explicit upper bound on the number of parts. An application of this lemma is also formalised, namely Roth’s theorem on arithmetic progressions in qualitative form via the triangle removal lemma.

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Mathematics of computing → Extremal graph theory
Keywords
  • Lean
  • formalisation
  • formal proof
  • graph theory
  • combinatorics
  • additive combinatorics
  • Szemerédi’s Regularity Lemma
  • Roth’s Theorem

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References

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