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UPPER BOUND THEOREM FOR ODD-DIMENSIONAL FLAG TRIANGULATIONS OF MANIFOLDS

Part of: Algebraic combinatorics Graph theory

Published online by Cambridge University Press:  01 June 2016

Michał Adamaszek  and
Jan Hladký
Michał Adamaszek
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark email aszek@mimuw.edu.pl
Jan Hladký
Affiliation:
Institute of Mathematics, Czech Academy of Science, Žıtná 25, 110 00, Praha, Czech Republic email honzahladky@gmail.com
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Abstract

We prove that among all flag triangulations of manifolds of odd dimension $2r-1$ , with a sufficient number of vertices, the unique maximizer of the entries of the $f$ -, $h$ -, $g$ - and $\unicode[STIX]{x1D6FE}$ -vector is the balanced join of $r$ cycles. Our proof uses methods from extremal graph theory.

MSC classification

Primary: 05E45: Combinatorial aspects of simplicial complexes 05C35: Extremal problems
Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 909 - 928
Copyright
Copyright © University College London 2016 

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References

Adamaszek, M., An upper bound theorem for a class of flag weak pseudomanifolds. Preprint, 2013,arXiv:1303.5603.Google Scholar
Adamaszek, M. and Hladký, J., Dense flag triangulations of 3-manifolds via extremal graph theory. Trans. Amer. Math. Soc. 367(4) 2015, 2743–2764.CrossRefGoogle Scholar
Aisbett, N., Frankl–Füredi–Kalai inequalities on the 𝛾-vectors of flag nestohedra. Discrete Comput. Geom. 51(2) 2014, 323–336.CrossRefGoogle Scholar
Charney, R., Metric geometry: connections with combinatorics. In Proceedings of FPSAC Conference (DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 24 ), American Mathematical Society (Providence, RI, 1996), 55–69.Google Scholar
Charney, R. and Davis, M., The Euler characteristic of a non-positively curved, piecewise Euclidean manifold. Pacific J. Math. 171 1995, 117–137.Google Scholar
Erdős, P., On some new inequalities concerning extremal properties of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press (New York, 1968), 77–81.Google Scholar
Erdős, P., Frankl, P. and Rödl, V., The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs Combin. 2(2) 1986, 113–121.Google Scholar
Flores, A., Über die Existenz n-dimensionaler Komplexe die nicht im den ℝ2n topologisch einbettar sind. Ergeb. Math. Kolloq. 5 1933, 17–24.Google Scholar
Fox, J., A new proof of the graph removal lemma. Ann. of Math. (2) 174(1) 2011, 561–579.Google Scholar
Frohmader, A., Face vectors of flag complexes. Israel J. Math. 164 2008, 153–164.Google Scholar
Gal, Ś. R., Real root conjecture fails for five- and higher-dimensional spheres. Discrete Comput. Geom. 34(2) 2005, 269–284.Google Scholar
Galewski, D. E. and Stern, R. J., Classification of simplicial triangulations of topological manifolds. Ann. of Math. (2) 111 1980, 1–34.CrossRefGoogle Scholar
Goodarzi, A., Convex hull of face vectors of colored complexes. European J. Combin 36 2014, 247–250.Google Scholar
Gromov, M., Hyperbolic groups. In Essays in Group Theory (ed. Gersten, S. M.), M.S.R.I. Publ. 8, Springer (1987), 75–264.Google Scholar
Karu, K., The cd-index of fans and posets. Compos. Math. 142(3) 2006, 701–718.CrossRefGoogle Scholar
Lutz, F. and Nevo, E., Stellar theory for flag complexes. Math. Scand. 118(1) 2016, 70–82.Google Scholar
Munkres, J. R., Topological results in combinatorics. Michigan Math. J. 31(1) 1984, 113–128.CrossRefGoogle Scholar
Murai, S. and Nevo, E., On the cd-index and 𝛾-vector of S*-shellable CW-spheres. Math. Z. 271(3–4) 2012, 1309–1319.CrossRefGoogle Scholar
Nevo, E. and Petersen, T. K., On 𝛾-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom. 45(3) 2011, 503–521.Google Scholar
Nevo, E., Petersen, T. K. and Tenner, B. E., The 𝛾-vector of a barycentric subdivision. J. Combin. Theory, Ser. A 118(4) 2011, 1364–1380.Google Scholar
Novik, I., Upper bound theorems for homology manifolds. Israel J. Math. 108(1) 1998, 45–82.Google Scholar
Ruzsa, I. Z. and Szemerédi, E., Triple systems with no six points carrying three triangles. In Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II (Colloq. Math. Soc. János Bolyai, 18 ), North-Holland (Amsterdam, New York, 1978), 939–945.Google Scholar
Simonovits, M., A method for solving extremal problems in graph theory, stability problems. In Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press (New York, 1968), 279–319.Google Scholar
Stanley, R. P., The upper bound conjecture and Cohen–Macaulay rings. Stud. Appl. Math. 54 1975, 135–142.Google Scholar
Stanley, R. P., Combinatorics and Commutative Algebra (Progress in Mathematics), Birkhäuser (Boston, 2004).Google Scholar
van Kampen, E. R., Komplexe in euklidischen räumen. Abh. Math. Semin. Univ. Hambg. 9 1932, 72–78.Google Scholar
Wagner, U., Minors in random and expanding hypergraphs. Proc. 27th Annual ACM Symposium on Computational Geometry (SoCG) 2011, 351–360.Google Scholar
Zheng, H., The flag upper bound theorem for 3- and 5-manifolds. Preprint, 2015, arXiv:1512.06958.Google Scholar
Zykov, A. A., On some properties of linear complexes. Mat. Sb. 24(66) 1949, 163–188.Google Scholar