Formalizing the solution to the cap set problem

Sander R. Dahmen, Johannes Hölzl, Robert Y. Lewis

Research output: Chapter in Book / Report / Conference proceedingConference contributionAcademicpeer-review

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In 2016, Ellenberg and Gijswijt established a new upper bound on the size of subsets of 픽nq with no three-term arithmetic progression. This problem has received much mathematical attention, particularly in the case q = 3, where it is commonly known as the cap set problem. Ellenberg and Gijswijt’s proof was published in the Annals of Mathematics and is noteworthy for its clever use of elementary methods. This paper describes a formalization of this proof in the Lean proof assistant, including both the general result in 픽nq and concrete values for the case q = 3. We faithfully follow the pen and paper argument to construct the bound. Our work shows that (some) modern mathematics is within the range of proof assistants.

Original languageEnglish
Title of host publication10th International Conference on Interactive Theorem Proving (ITP 2019)
EditorsJohn Harrison, John O'Leary, Andrew Tolmach
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages19
ISBN (Print)9783959771221
Publication statusPublished - 2019
Event10th International Conference on Interactive Theorem Proving, ITP 2019 - Portland, United States
Duration: 9 Sept 201912 Sept 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference10th International Conference on Interactive Theorem Proving, ITP 2019
Country/TerritoryUnited States


  • Cap set problem
  • Combinatorics
  • Formal proof
  • Lean


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