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UPPER BOUND THEOREM FOR ODD-DIMENSIONAL FLAG TRIANGULATIONS OF MANIFOLDS

  • Michał Adamaszek (a1) and Jan Hladký (a2)

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.

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1. Adamaszek, M., An upper bound theorem for a class of flag weak pseudomanifolds. Preprint, 2013,arXiv:1303.5603.
2. Adamaszek, M. and Hladký, J., Dense flag triangulations of 3-manifolds via extremal graph theory. Trans. Amer. Math. Soc. 367(4) 2015, 27432764.
3. Aisbett, N., Frankl–Füredi–Kalai inequalities on the 𝛾-vectors of flag nestohedra. Discrete Comput. Geom. 51(2) 2014, 323336.
4. 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), 5569.
5. Charney, R. and Davis, M., The Euler characteristic of a non-positively curved, piecewise Euclidean manifold. Pacific J. Math. 171 1995, 117137.
6. Erdős, P., On some new inequalities concerning extremal properties of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press (New York, 1968), 7781.
7. 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, 113121.
8. Flores, A., Über die Existenz n-dimensionaler Komplexe die nicht im den ℝ2n topologisch einbettar sind. Ergeb. Math. Kolloq. 5 1933, 1724.
9. Fox, J., A new proof of the graph removal lemma. Ann. of Math. (2) 174(1) 2011, 561579.
10. Frohmader, A., Face vectors of flag complexes. Israel J. Math. 164 2008, 153164.
11. Gal, Ś. R., Real root conjecture fails for five- and higher-dimensional spheres. Discrete Comput. Geom. 34(2) 2005, 269284.
12. Galewski, D. E. and Stern, R. J., Classification of simplicial triangulations of topological manifolds. Ann. of Math. (2) 111 1980, 134.
13. Goodarzi, A., Convex hull of face vectors of colored complexes. European J. Combin 36 2014, 247250.
14. Gromov, M., Hyperbolic groups. In Essays in Group Theory (ed. Gersten, S. M.), M.S.R.I. Publ. 8, Springer (1987), 75264.
15. Karu, K., The cd-index of fans and posets. Compos. Math. 142(3) 2006, 701718.
16. Lutz, F. and Nevo, E., Stellar theory for flag complexes. Math. Scand. 118(1) 2016, 7082.
17. Munkres, J. R., Topological results in combinatorics. Michigan Math. J. 31(1) 1984, 113128.
18. Murai, S. and Nevo, E., On the cd-index and 𝛾-vector of S*-shellable CW-spheres. Math. Z. 271(3–4) 2012, 13091319.
19. Nevo, E. and Petersen, T. K., On 𝛾-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom. 45(3) 2011, 503521.
20. Nevo, E., Petersen, T. K. and Tenner, B. E., The 𝛾-vector of a barycentric subdivision. J. Combin. Theory, Ser. A 118(4) 2011, 13641380.
21. Novik, I., Upper bound theorems for homology manifolds. Israel J. Math. 108(1) 1998, 4582.
22. 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), 939945.
23. 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), 279319.
24. Stanley, R. P., The upper bound conjecture and Cohen–Macaulay rings. Stud. Appl. Math. 54 1975, 135142.
25. Stanley, R. P., Combinatorics and Commutative Algebra (Progress in Mathematics), Birkhäuser (Boston, 2004).
26. van Kampen, E. R., Komplexe in euklidischen räumen. Abh. Math. Semin. Univ. Hambg. 9 1932, 7278.
27. Wagner, U., Minors in random and expanding hypergraphs. Proc. 27th Annual ACM Symposium on Computational Geometry (SoCG) 2011, 351360.
28. Zheng, H., The flag upper bound theorem for 3- and 5-manifolds. Preprint, 2015, arXiv:1512.06958.
29. Zykov, A. A., On some properties of linear complexes. Mat. Sb. 24(66) 1949, 163188.
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