research-article

The Resolution of Keller’s Conjecture

Authors:

Joshua Brakensiek,

David NarváezAuthors Info & Claims

Journal of Automated Reasoning, Volume 66, Issue 3

Pages 277 - 300

https://doi.org/10.1007/s10817-022-09623-5

Published: 01 August 2022 Publication History

Abstract

We consider three graphs,

G_{7, 3}

,

G_{7, 4}

, and

G_{7, 6}

, related to Keller’s conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size

2^{7} = 128

. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of

R^{7}

contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of

R^{8}

exists (which we also verify), this completely resolves Keller’s conjecture.

References

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Cited By

Heule MScheucher M(2024)Happy Ending: An Empty Hexagon in Every Set of 30 PointsTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-031-57246-3_5(61-80)Online publication date: 6-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57246-3_5
Clune JKrebbers RTraytel DPientka BZdancewic S(2023)A Formalized Reduction of Keller’s ConjectureProceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3573105.3575669(90-101)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3573105.3575669

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Published In

cover image Journal of Automated Reasoning

Journal of Automated Reasoning Volume 66, Issue 3

Aug 2022

158 pages

ISSN:0168-7433

Issue’s Table of Contents

© The Author(s), under exclusive licence to Springer Nature B.V. 2022.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 August 2022

Accepted: 22 February 2022

Received: 23 April 2021

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Cited By

Heule MScheucher M(2024)Happy Ending: An Empty Hexagon in Every Set of 30 PointsTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-031-57246-3_5(61-80)Online publication date: 6-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57246-3_5
Clune JKrebbers RTraytel DPientka BZdancewic S(2023)A Formalized Reduction of Keller’s ConjectureProceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3573105.3575669(90-101)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3573105.3575669

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