  A \emph{stacked triangulation} of a $d$-simplex $\mathbf{o}=\{1,\ldots,d+1\}$
($d\geq 2$) is a triangulation obtained by repeatedly subdividing a $d$-simplex
into $d+1$ new ones via a new vertex (the case $d=2$ is known as an Appolonian
network). We study the occurrence of such a triangulation in the
Linial--Meshulam model, i.e., for which $p$ does the random simplicial complex
$Y\sim \mathcal{Y}_d(n,p)$ contain the faces of a stacked triangulation of the
$d$-simplex $\mathbf{o}$, with its internal vertices labeled in $[n]$. In the
language of bootstrap percolation in hypergraphs, it pertains to the threshold
for $K_{d+2}^{d+1}$, the $(d+1)$-uniform clique on $d+2$ vertices.
  Our main result identifies this threshold for every $d\geq 2$, showing it is
asymptotically $(\alpha_d n)^{-1/d}$, where $\alpha_d$ is the growth rate of
the Fuss--Catalan numbers of order $d$. The proof hinges on a second moment
argument in the supercritical regime, and on Kalai's algebraic shifting in the
subcritical regime.
