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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Dense flag triangulations of $3$-manifolds via extremal graph theory

Authors: Michał Adamaszek and Jan Hladký
Journal: Trans. Amer. Math. Soc. 367 (2015), 2743-2764
MSC (2010): Primary 05E45; Secondary 05A15
Published electronically: August 8, 2014
MathSciNet review: 3301880
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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

Jan Hladký
Affiliation: Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL, United Kingdom

Keywords: $f$-vector, simplicial complex, Gal’s conjecture, flag triangulations of $3$-manifolds
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.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.