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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Dense flag triangulations of $3$-manifolds via extremal graph theory
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by Michał Adamaszek and Jan Hladký PDF
Trans. Amer. Math. Soc. 367 (2015), 2743-2764 Request permission

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