The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h -vector.
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Hersh, Novik A Short Simplicial h -Vector and the Upper Bound Theorem. Discrete Comput Geom 28, 283–289 (2002). https://doi.org/10.1007/s00454-002-0746-7