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The Dirac–Goodman–Pollack Conjecture.

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Title: The Dirac–Goodman–Pollack Conjecture.
Authors: Dumitrescu, Adrian1 (AUTHOR) ad.dumitrescu@algoresearch.org
Source: Discrete & Computational Geometry. Sep2024, Vol. 72 Issue 2, p728-742. 15p.
Subjects: Discrete geometry, Logical prediction, Generalization, Permutations
Abstract: In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of n noncollinear points in the plane has a point incident to at least cn connecting lines determined by the set. The notion of allowable sequences of permutations provides a natural combinatorial setting for analyzing these problems. Within this formalism, the conjectured generalization reads as follows: Any nontrivial allowablen-sequence Σ has a local sequence Λ i whose half-period is at leastcn. The conjecture is confirmed here with a concrete bound c = 1 / 845 . Several related problems are discussed. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: The Dirac–Goodman–Pollack Conjecture.
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  Data: In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of n noncollinear points in the plane has a point incident to at least cn connecting lines determined by the set. The notion of allowable sequences of permutations provides a natural combinatorial setting for analyzing these problems. Within this formalism, the conjectured generalization reads as follows: Any nontrivial allowablen-sequence Σ has a local sequence Λ i whose half-period is at leastcn. The conjecture is confirmed here with a concrete bound c = 1 / 845 . Several related problems are discussed. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s00454-023-00487-z
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      – TitleFull: The Dirac–Goodman–Pollack Conjecture.
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              Text: Sep2024
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