Superpolynomial lower bounds for vertex numbers of real projective space triangulations via a topological Figiel-Lindenstrauss-Milman theorem

We prove that every simplicial triangulation of real projective $d$-space has $\exp (\mathrm{\Omega }(\sqrt{d}))$ vertices. Together with known constructions, this determines the minimum vertex number as ${\mu }_d=\exp (d^{1\mathrm{/}2+o(1)})$. The result follows from a topological generalization of the Figiel–Lindenstrauss–Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: a finite strongly regular CW complex with a free cellular involution, $v$ vertices, and $f$ maximal cells has $\mathbb{Z}\mathrm{/}2$-index at most $O(\log v\log f)$. We bound the dimensions of Morse cells by a trace estimate for a constrained Hessian, obtaining a Morse-theoretic proof of the classical inequality for centrally symmetric polytopes. As further applications of this inequality, we give an $\exp (\mathrm{\Omega }(\sqrt{t}))$ lower bound for the order of a triangle-free topologically $t$-chromatic graph and bound the index of sign complexes by $O(d{\log }^{2}N)$ for total matrices and $O(d{\log }^{3}N)$ for partial matrices, where $N\ge 2$ is the number of columns and $d\ge 1$ is the VC dimension.