Remarks on the upper bound theorem. (Q1417958)

An extension of the known upper bound theorem for convex polytopes in \(\mathbb{R}^d\) is the so-called strong upper bound theorem due to Gil Kalai, holding for the boundary complexes of simplicial polytopes and their full-dimensional subcomplexes. The author gives an extension of this theorem for even-dimensional simplicial manifolds. As a second result, the paper contains an upper bound theorem for odd-dimensional, centrally symmetric simplicial complexes (i.e., simplicial complexes for which there exists an involution \(\psi: K\to K\) that fixes none of their faces).