Fix a prime $p$. What is the smallest integer $n$ so that there is a simplicial complex on $n$ vertices with $p$-torsion in its homology?

For example, when $p=2$, there is a complex with 6 vertices (the minimal triangulation of the real projective plane) with 2-torsion in its homology. I'm pretty sure that it's the smallest possible: with 5 or fewer vertices, there should be no torsion at all. When $p=3$, there is a complex with 9 vertices (a triangulation of the mod 3 Moore space, for instance) with 3-torsion. Is there one with 8 vertices? With $p=5$, there is a complex with 11 vertices, found by randomly testing such complexes on my computer.

We can refine this: fix $p$ and also a positive integer $d$. What's the smallest $n$ so that there is a simplicial complex $K$ on $n$ vertices with $p$-torsion in $H_d(K)$? Or we can turn it around: for fixed $n$, what kinds of torsion can there be in a simplicial complex on $n$ vertices?

(A paper by Soulé ("Perfect forms and the Vandiver conjecture") quotes a result by Gabber which leads to a bound on the size of the torsion for a fixed number $n$ of vertices; however, this bound is far from optimal, at least for small $n$.)