(This should be a comment, but it is too long to be comfortably edited there)

According to a theorem of Arnoux and Marin [Arnoux, Pierre; Marin, Alexis. The Kühnel triangulation of the complex projective plane from the view point of complex crystallography. II. Mem. Fac. Sci. Kyushu Univ. Ser. A 45 (1991), no. 2, 167--244. [MR1133113](http://www.ams.org/mathscinet-getitem?mr=MR1133113))], a triangulation of $\mathbb RP^d$ has at least $(d+1)(d+2)/2$ vertices. Triangulations were explicitely constructucted by Kühnel [Kühnel, W. Minimal triangulations of Kummer varieties. Abh. Math. Sem. Univ. Hamburg 57 (1987), 7--20. [MR0927159](http://www.ams.org/mathscinet-getitem?mr=MR0927159), with $2^{d+1}-1$ vertices and $\frac12(d+1)!$ simplices. 

Having lower bounds for the number of vertices $n$, makes it useful to know that there exist lower and upper bounds for the $f$-vectors (*i.e.*, for the number of simplices) in terms of $d$ and $n$. Two nice ones are those of Kalai and Gromov for a lower bound ([MR0877009](http://www.ams.org/mathscinet-getitem?mr=MR0877009))], and of Novik for an upper bound ([MR1669325](http://www.ams.org/mathscinet-getitem?mr=MR1669325)). 

Our fellow MOer Gil could probably tell us about the state of the art :)