  This paper concerns the facial geometry of the set of $n \times n$
correlation matrices. The main result states that almost every set of $r$
vertices generates a simplicial face, provided that $r \leq \sqrt{\mathrm{c}
n}$, where $\mathrm{c}$ is an absolute constant. This bound is qualitatively
sharp because the set of correlation matrices has no simplicial face generated
by more than $\sqrt{2n}$ vertices.
