Dense flag triangulations of $3$-manifolds via extremal graph theory
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- by Michał Adamaszek and Jan Hladký PDF
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Abstract:
We characterize $f$-vectors of sufficiently large three-dimensional flag Gorenstein$^*$ complexes, essentially confirming a conjecture of Gal [Discrete Comput. Geom., 34 (2), 269–284, 2005]. In particular, this characterizes $f$-vectors of large flag triangulations of the $3$-sphere. Actually, our main result is more general and describes the structure of closed flag 3-manifolds which have many edges.
Looking at the 1-skeleta of these manifolds we reduce the problem to a certain question in extremal graph theory. We then resolve this question by employing the Supersaturation Theorem of Erdős and Simonovits.
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Additional Information
- Michał Adamaszek
- Affiliation: Fachbereich Mathematik, Universität Bremen, Bibliothekstr. 1, 28359 Bremen, Germany
- Address at time of publication: Max Planck Institute for Informatics, Campus E1 4, 66123 Saarbrücken, Germany
- Email: aszek@mimuw.edu.pl
- Jan Hladký
- Affiliation: Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL, United Kingdom
- Email: honzahladky@gmail.com
- Received by editor(s): November 29, 2012
- Received by editor(s) in revised form: April 11, 2013
- Published electronically: August 8, 2014
- Additional Notes: The research of the first author was carried out while he was a member of the Centre for Discrete Mathematics and its Applications (DIMAP), supported by the EPSRC award EP/D063191/1.
The second author is an EPSRC Research Fellow. - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 2743-2764
- MSC (2010): Primary 05E45; Secondary 05A15
- DOI: https://doi.org/10.1090/S0002-9947-2014-06153-9
- MathSciNet review: 3301880