Geodesic
The flag upper bound theorem for 3- and 5-manifolds
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020).

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We prove that among all flag 3-manifolds on n vertices, the join of two circles with [n 2] and [n 2] vertices respectively is the unique maximizer of the face numbers. This solves the first case of a conjecture due to Lutz and Nevo. Further, we establish a sharp upper bound on the number of edges of flag 5-manifolds and characterize the cases of equality. We also show that the inequality part of the flag upper bound conjecture continues to hold for all flag 3-dimensional Eulerian complexes and characterize the cases of equality in this class. 
@article{DMTCS_2020_special_379_a17,
     author = {Zheng, Hailun},
     title = {The flag upper bound theorem for 3- and 5-manifolds},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     year = {2020},
     doi = {10.46298/dmtcs.6335},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6335/}
}
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Zheng, Hailun. The flag upper bound theorem for 3- and 5-manifolds. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi : 10.46298/dmtcs.6335. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6335/

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